First, you have to solve Mudhenge in The Big Moist, through trial and error. Turn off the music volume for this one, but keep the effects volume on. Go through the leftmost gate first, and pay attention to the sound. That sound means you've gone through the correct gate. Keep doing that until you reach the violent nudists, which is the final destination of mudhenge.

Or if you're too lazy, the solution is 1, 2, 1, 2, 5, 2, 5, 4, 3, each gating being labelled from 1-5, with the leftmost gate being 1.

Fight off the nudists (taking off your equipment also bypasses the nudist fight), and place your compass on the altar. Think real hard about meat, then go through the west-north gate (don't worry, you can't go through the wrong gate here). You'll arrive at a riverboat. You need to reach the top of the riverboat- first, talk to the speaker to get a code that turns off the steam.

Again, if you're too lazy to figure out the puzzle, the solution is 1-4-0-0-5-4.
The outhouse combination is 04 04 54 (thank you Sublime Hamster)

The rest is straightforward, but when you're done and you've finished with Captain Dan, return to Mudhenge. Place the compass on the altar again and think even harder about meat.
Go through the south gate, and you'll be inside the vault!
Exit, and you'll get the achievement.

Outhouse code was changed in the most recent patch update !

If you go to the Hardware store at Crystaldream lake and interact with the barrels, you'll notice that one is on sale for 10,000 meat! This is the only barrel you can buy in the game, so tough luck buttercup, you gotta get rich and get rich quick.
You can likely stockpile a good few thousand meat just by playing the game, but if you want to make this easier, pick up meat drop perks and items for a while, especially the Shadow Coin (requires high shadow taint, see my other guide for more info) and shadow mead which give huge boosts to meat drops.
Additionally, there's silver bars hidden throughout the game which sell for 1000 meat!
There's one in the water under the Dauphin house in The Big Moist, and one underneath the Mob Submarine (both require the diving helmet), and others to find. You should also pick up several valuable items like old engagement rings as you play through the game which sell for a few hundred meat or so, so loot as many jewelry boxes as possible throughout your playthrough.

But basically, this is an end-game buy, because chances are without some prep work you're gonna end up selling a lot of your stuff.

Buying the barrel nets you the achievement and a fishing spot for infinite miscellaneous hardware junk like matches, fuses, anarchist's hardware, etc., and a portable barrel accessory which gives +7 physical armor...yay.

For this one, it's better to wait until the last day, but you're going to want the following:

• SAVE the bathtub found in the junkyard until you're ready to get the achievement, it grants you a +1 in each stat after taking a bath and makes this way easier
  
• A food item which boosts each stat by 1 (either the pressed fruit found in junk mail or the fruit cocktail found in the Dauphin house)
  
• 4 "crackers" from the Government Valley prison
  
• The prologue cat unlocked, as their boon gives +1 to each stat
  
• The glowy orb found in the radio shack in The Big Moist, gives a +1 to mysticality when you ponder it in your room
  
• Max upgraded stats (obviously)
  
• The pants shop and/or potion shop, if possible
  
• The statue the puckwudgies erect after you terrorize their town
  
• Absinthe, Whiskey, and Tequila unlocked in the speakeasy- even though you can only have one drink effect at a time, it's better to have all three so you can pick which one you need, as each grants a +1 to two separate stats
  
• A potion that boosts two separate stats, such as the radium rouge found in the Dauphin house
  
• Complete monster sash- it's one of the best accessories in the game plus it gives you +1 to each stat if you get all badges (you don't need all the badges, just the ones that boost stats, but it's better to finish it if you can)
  
• Grub hat from Haim Quarry. I know it's disgusting, but it gives +1 to all stats some-♥♥♥♥♥♥♥ how. You get it by putting both eyes in the statue and then having the narrator go into detail about how you destroy the grubs in the fly cave
  
• You can also get a +1 mysticality by having your fortune told on the boardwalk 3 times

Got all that? Good. Here's the breakdown of getting your 15 stats across the board:
After fully upgrading your stats and taking the prologue cat's boon (it doesn't have a set name), you should be at 6 in every stat.
Take the food item you stashed, and the bath if it's still available to get you up to 8.
Your sash and disgusting grub hat will push you up to 10.
Now, trade in the crackers at the prison for four separate perks- talk to the guy who's whistling, the guy floating an orange, the chiropractor, and the guy lifting weights to push your moxie and mysticality up to 11, and your muscle to 12.
If you've gotten this far as described, it would be best to take a shot of Absinthe to get have all stats be 12.
Now wander around the map, get into fights, and lose in them to get the effects from losing which boost your stats until all your stats are 13.
Ponder the glowy orb to raise your mysticality- you're now 13/14/13 across the board.
For the final stats, you'll have to combine your pants and potion effects to cover the rest. This is why the pants shop is recommended, as they sell a pair of pants which raises two separate stats by one. The potions shop is also recommended as it is possible to buy a potion raising all stats by one.
The reason you want the puckwudgie statue is because if you happen to be missing a point in a stat, but the others are 15, you can interact with it to gain that final point.

When all is said and done, go to the boardwalk and rock the carnival man's world.
Grab that achievement like the hardcore gamer you are.

In the Goldwaith Park, there are some giant crickets in the Butterfly Zoo. Notice how the prompt mentions you can "hear a faint chirping in your mind"- return to them when you have 9 mysticality (on normal difficulty), and you'll be able to talk to them.
Listen to their requests, and if you put the crystal they give you in the portal at the end of the game, you get the achievement.
(Thank you DineAtColombine)

Take Chemistry at S.I.T- while there, you're going to want to make a barnacle dissolving solution (1-4-4-4), and save it for later. Then, progress the Mob questline until you unlock the Mob Submarine on the map. If you haven't already, tear the diver's helmet off the statue outside Fission Chips (high strength check), and bring it with you to the submarine. Fix the pressure in the submarine, and go underwater using your diving helmet. Use the barnacle dissolver on the chest, and you get the achievement + a necklace which gives you +7 to every elemental armor.
(thank you DineAtColombine)

When you project yourself into the pocketwatch's curse, do not try to eat the cat- leave it alone. Interact with the objects in the room (including shaking hands with Charles and signing Jessica's book). Once you've raised your dexterity to 7, you'll be able to pet the cat, giving you a total of 11 dexterity if you've successfully interacted with every object in the room. Do not open the door. Instead walk up to the radio in the corner of the room and tune it (which requires 11 dexterity), then you'll get the achievement.
(Thank you DineAtColombine)