Disclaimer: I am no expert, this is just what I do or what somebody I know does.
I would recommend taking a Level 30 Warrior to the High Rift but you can do it with others. There are lots of times that I don't fight the monsters and the Warrior is best at just taking the hits. Despite what I recommend, I use my Level 30 Rogue but the people I play with use their Warriors. I wear the High Rift Rogue armor for the double jump pants which once I remembered to use it has come in mighty handy. If you take a warrior you could use the cape of wings to get the same effect as double jump. You can have all these great stats and armor but I think it comes down to how familiar you are with the rooms and how fast you can get through them.
This is what I keep on my hotbar. The order doesn't really matter

• Weapon
  
• Scroll of Teleportation
  
• Rocket
  
• Scroll of Rift Stability
  
• Explosive Bomb
  
• Mighty Healing Potion
  
• Super Drill or Wisp in a Bottle III or Scroll of Revelation
  
• Dirt Block

1. Weapon - to kill monsters obviously
   
2. Scroll of Teleportation - really helps speed of the rooms the require jumping across platforms
   
3. Rocket - some rooms required this, I use it all the time to get to chests
   
4. Explosive Bombs - fast way to find hidden chest and get crystals from walls
   
5. Mighty Healing Potion - I don't know about you but I always end of taking damage from monsters and traps
   
6. Scroll of Rift Stability - for those moments when you need a few seconds to get to the floor exit

1. Dirt Block - I always always carry dirt, but the only time I use it in the rift it to block wall traps, very handy
   
2. Drill or Pick - some chest you have to break blocks to get to but I use an Explosive Bomb for that (bombs are easier and faster except in the mines where there is the danger of destroying the floor)
   
3. Wisp in a Bottle - there is a darkness room that this is great for and this is really helpful when trying to spot the little holes in the wall for hidden chest (I don't use it much for either)
   
4. Scroll of Revelation - there are a few rooms that require this but there are chest that provide them (I switch is to my hotbar just for those rooms)

For each level you have close to 13 minutes and that is with getting all the recover rift stability cubes. I timed out how long you have to do the first floor and it was about 7 minutes (playing solo). There are recover rift stability cubes all over the floor and in each room. When you activate an obelisk you also receive to rift stability cubes. The scrolls of rift stability have a 300 (5 minute) cooldown so you can only use one per floor and it doesn't add much time. I only use them last minute when I am desperate. So you really need to do each room in under 2 minutes to give you time to get around the floor and find the room teleport pads.

• If you do not need to kill monsters to unlock a door don't fight them. I just take the hits and keep moving unless I have to stay in one place for long and there are enough to kill me then I fight.
  
• Rockets can be very useful for getting over wall or up on roofs
  
• Double jump pants really help when trying to avoid the light bombs or fireballs coming at you if you play as a rogue
  
• Try to get the chest as you go through so you don't have to back track. I can usually complete a room in about a minute by already knowing where the chests are and just making a quick loop through the room.

• If you see dirt, stone, or concrete in the walls break it (explosive bomb). The blocks that make of most of the rift buildings are unbreakable so when you throw a bomb at the wall only the dirt/stone/concrete will break. I find a lot of chest this way. Same goes for floors but be careful not to fall through.
  
• There are lots of hidden on the outside of buildings. Look around corners, holes in walls, and windows for a little ledge. You may not always see a chest but if there are blocks you can jump to there could be a chest at the end of the path (teleportation scrolls can help with this).
  
• The chest make a noise just like the crystal spiders, so you can here when one is close by.
  
• If you have trouble memorizing where chest are, I would try to follow a circular path through the room checking for chest along the way to getting the obelisk. Wisp in a Bottle can help you spot holes in walls and other clues.

There are a series of buttons that open a series of doors to unlock obelisk.
Chest 1: to the right of the yellow button in hole in wall hole up small path

Chest 2: behind a secret door bookcase in one of the rooms


There are four green buttons hidden in the sand along the walls (two on each side) that need to be pushed to unlock the obelisk.
Chest 1: under the bridge, buried in the sand

Chest 2: outside the wall on the left


There are two green buttons (one in each tower) that need to be pushed to unlock the obelisk.
Chest 1: to the left of one of the green buttons

Chest 2: above the obelisk. The doorway in on the left side of the roof. There is a way to jump up or you could use a rocket.


Kill the blossoms, I would leave the traps on to help. Also the blue switch needs to be on the get chest 2. Then you go downstairs and there are pressure plates the you have to jump across. Since there are no green traps along the wall, jump across following the green pressure plates.
Chest 1: hidden among the pressure plates. There are two green ones together with a little platform in between. Jump down and you will see a little opening in the wall. Outside is the chest.

Chest 2: up the right stair at the end, behind blue door


There is a green button to the right through the fire traps, the green door will unlock. Kill all the monsters inside to unlock obelisk.
Chest 1: buried in the dirt to the right of the green button

Chest 2: on the roof of the building, path up is to the right or use rocket


There are four red buttons among the walls (two to the left, one under obelisk, one to the right) that will unlock the obelisk. You don't have to fight the monsters.
Chest 1: outside the front right corner

Chest 2: outside the back left corner


The obelisk is downstairs so, run through the house, go downstairs, and activate the obelisk. Kill the monsters that spawned and unlock the shortcut back to the teleport pad.
Chest 1: on the roof

Chest 2: outside the staircase (rocket over wall, jump cupboard, or use path from mid stairs)


The room/maze is dark until obelisk is activated. You can use flares, wisp in a bottle, or just run around in the dark. Kill all monsters (four hollow knights) to unlock obelisk.
Chest 1: hidden in the wall, hole is a corner of same wall as obelisk

Chest 2: behind a red door, push the two red buttons to unlock


Use rockets to get around and destroy the monster spawners.
Chest 1: on roof of some tower

Chest 2: behind wall near stairs leading to obelisk


Jump across to get to obelisk
Chest 1: outside of right wall closer to obelisk

Chest 2: on the ground near teleport pad

Jump across platforms while avoiding traps to get to obelisk (I used teleportation scroll to get to the middle then the end)
Chest 1: in the tower to the left (use teleportation scroll)

Chest 2: above the obelisk, way up is next to the trap or use rocket


Kill the red skeletons and the red platforms will become solid. Kill all monsters in the building to unlock obelisk.
Chest 1: under obelisk, the is a white platform path to get the use scroll of revelation or teleportation scroll

Chest 2: under the small platform to the left (or on your return)


Push the two green buttons (up each stairs and down hall) to unlock obelisk.
Chest 1: above obelisk, jump up right side or use rocket

Chest 2: hidden in wall where first trap hits wall


Push the two green buttons on the roof (staircase on back of building) between each set of the coffins to unlock the green door. Then kill the monster inside to unlock obelisk.
Chest 1: under floor near staircase

Chest 2: on the roof


Use scrolls of revelation to move from island to island. The obelisk is at the top.
Chest 1: on an island

Chest 2: on an island


Go around the building til you can enter the courtyard, there are four white buttons. Enter the right combination, the buttons stay lit as they are pushed in the correct order. Combination - 2413 (from left to right)
Chest 1: to the right of teleport pad

Chest 2: there is a hole in the water near the boardwalk, chest down hole


Destroy the monster spawner in center building to unlock obelisk.
Chest 1: in building to the right of teleport pad

Chest 2: push two blue buttons to unlock, first button left of teleport pad, second behind building at end of path (farthest from teleport pad)


Go through and kill all the monster to unlock the obelisk.
Chest 1: green button in first room through hole in corner

Chest 2: under floor in front of green door, break dirt in corner


Push the two green buttons each around the corner to the left and right of teleport pad.
Chest 1: red button down hallway behind obelisk

Chest 2: outside near red button


You need to go underwater to (I go to the left to make a circle) find the yellow button behind the stairs that led into the water (first chest is near). Keep going around to a second stairs, this room should have a monster spawner. Obelisk unlocks when you destroy it.
Chest 1: underwater near yellow button go up the stairs

Chest 2: behind room with the monster spawner

This floor is a single room with three obelisk. Open the chest to get the explosive bombs then start using the bombs to break through the dirt to get to the "basement". Once down there, throw a bomb at any group of dirt you see till you break through (usually one bomb is enough). Go through all passages, obelisk are fairly easy to spot.
Chest 1: behind the first large wall of dirt to the left, there is a second wall of dirt at the end of the hallway

Chest 2: in the water behind a wall of dirt near an obelisk

Find your way through the library to the blue button to unlock the obelisk. You can place dirt blocks in front of traps to block them (just make sure you put them where the trap is and not just where the light bomb hits)
Chest 1: behind secret door bookcase

Chest 2: behind obelisk, break concrete blocks


Work your way through the mine to get to the obelisk. You will need to kill monsters along the way, focus on the casters to defeat them quicker.
Chest 1: behind red door, one button in each area with the poison

Chest 2: in second poison area to the left behind dirt


Defeat the monsters inside to unlock obelisk. I get the yellow button outside first so I can fight all the monsters at once, then get the chest inside.
Chest 1: outside window is a yellow button, chest inside building

Chest 2: on the roof, you can jump up tree if needed, I use rocket


Go down into the crypt. You will need to find and kill the two skeletons to unlock the obelisk. The two white buttons for the chest on the surface are hidden behind the catacombs.
Chest 1: behind the stone on the right in the room with the skeleton

Chest 2: in building on surface behind white door, buttons are in the catacombs


Go straight across the platforms to the far building. There will be spiders (no need to kill them) and a yellow button. The button unlocks the obelisk. Go back out the just across to the obelisk.
Chest 1: in the middle of the water

Chest 2: under the obelisk, entrance in water between teleport pad and obelisk


Make your way through the hallway avoiding traps as best you can and getting the buttons for one chest along the way. You will come to a room with a monster, rotating traps, and two blue buttons. The blue buttons unlock the obelisk.
Chest 1: through hole in wall in room with rotating traps

Chest 2: behind locked white door next to obelisk, buttons in first hallway


This is a three story building with two yellow buttons, one on each top and bottom floor. Go to the right to go upstairs first. Get the button and chest up there first, then you can drop down through the chest room to get the second button.
Chest 1: hole in wall next to last trap on the left

Chest 2: in room under dirt behind the stairs leading to the top


Kill all the spores while avoiding traps to unlock obelisk. Note there are 3 chest here.
Chest 1: outside left of path near top

Chest 2: outside behind obelisk

Chest 3: under a pile of dirt


There is a button across the way that will unlock the monsters you need to kill to unlock the obelisk.
Chest 1: to the right of where the monsters came out there is a path along the outside where you will find a wall of dirt, break it and go in

Chest 2: under where the button was to the right



Chest 1:

Chest 2:

There are four torches on the house and four matching colored switches. Turn the switches on to match the torches on the house.
Chest 1: to the left side of the house

Chest 2: out the window behind the obelisk

There is a switch between two doors the changes which door is open. There is one yellow button and one chest behind and locked door in each room.
Chest 1: red button under dirt on left stairs

Chest 2: green button outside to the right of the rooms

Chest 1:

Chest 2:
Must destroy the monster spawners in the two caged structures to unlock obelisk. Use the traps to destroy them, one pressure plate turns trap the other turns it on for awhile.
Chest 1: under dirt of stairs near obelisk

Chest 2: under the monster spawner to the left

There are two hallways on light bombs traps you must get through to reach the obelisk. Good luck
Chest 1: under teleport pad where the first set of light bombs hit

Chest 2: in a hole in the wall to the left of the second set of traps

There are two sets of fire traps you must jump across to get to the obelisk.
Chest 1: behind the stairs after first fire trap

Chest 2: above the obelisk

You must kill all the monsters to unlock obelisk.
Chest 1: behind red door; one red button behind green door, green button at the top of one set of stairs, second red button along right ledge above trap area

Chest 2: hole in wall along left ledge above trap area leads to room above the obelisk

There are a yellow and green button on either side of the red door. Each red button is behind and wall of dirt in each room closest to the teleport pad, one behind a green door, one yellow.

Chest 1: hole in wall behind dirt in other yellow room

Chest 2: to the right of teleport pad around corner

Chest 1:
Chest 2:
Chest 1:
Chest 2:

There are two obelisk on this floor both behind monster door. After you jump down, I turned right. Kill all the water monsters including the ones in the water under the stairs, unlock first obelisk. Keep going around and kill all the crabs to unlock second obelisk. Exit will open, keep going around to the beginning. Kill the krakens to unlock exit.
Chest 1: on left ledge where you dropped from teleport pad

Chest 2: through drain tunnel where crabs were near obelisk

Chest 1:

Chest 2:

Chest 1:

Chest 2:

Chest 1:
Chest 2:
Chest 1:

Chest 2:

Chest 1:

Chest 2:

Chest 1:

Chest 2:
Chest 1:

Chest 2:
You have to push all four buttons and get to the obelisk before the first button times out. There are traps along each set of stairs between button that turn on and off, if you time it right you can go up the stairs while the traps are off.
Chest 1: on ledge under traps where you jump down to the obelisk

Chest 2: behind blue door, I haven't found the button yet, again where you drop down to obelisk

Chest 1:
Chest 2:
Chest 1:
Chest 2:

Chest 1:

Chest 2:



Chest 1:

Chest 2:


Kill all the blossoms as you go up the tower. Obelisk is at the top of the tower.

Chest 1: At the top of the stairs outside

Chest 2: Behind a tree there is a hole in the wall with a green button outside, green door is at bottom of stairs across from teleport pad


This room is hard to describe. There are fires everywhere, you could use water bombs to put the out, I just tried to avoid them. Eventually I came to a room where I had to kill the monsters to unlock the obelisk.
Chest 1: under the floor, down a small stairs somewhere along the way

Chest 2: under the room where you fight the monsters before the obelisk


The buttons that unlock the doors only stay on so long, so you will want to push them in the same order that you go through the doors.
Chest 1:

Chest 2:

Chest 1:
Chest 2:

Chest 1:
Chest 2:

Chest 1:
Chest 2:

Chest 1:
Chest 2:

Chest 1:
Chest 2:

This is the final boss floor. You need to activate two obelisk to unlock the boss fight. Kill the monsters to unlock the obelisk.
Chest 1: behind yellow door next to obelisk on the left, button under dirt in area with the monsters

Chest 2: behind the obelisk to the right