This is not a full step by step walkthrough of the DLC. However there are a few things worth knowing before you start, if you intend to get all the achievements. So I thought I'd do just a rundown on the achievement requirements, and how to avoid missing any of them.

Firstly: The new DLC is accessed via The Observatory in the base game. You need to have at least 50% data, and a new NPC will show up right near the breach in the middle of The Observatory. This NPC will give you an item, which you can activate at the breach there to put you in an alternate dimension called 'Blue Sun'. This will unlock the following achievement:

Serenity
Enter the Blue Sun universe


For achievement purposes, you will also want to be in a playthrough where The Elder Gods (Defas Nemundis, Ozyormy Goija and Uthos) from the base game have not been killed. As this is required for a later achievement. If you have already killed them, you can defeat the final boss of the game again, and go in to NG+. You can also reduce the difficulty back down to what it previously was with a fearful effigy, this will not affect any achievements.

Now, you might have noticed that not a whole lot has changed after entering the Blue Sun universe. The world is mostly the same, and you need to go find the new DLC areas.

Gehenna Prison can be accessed via Port Issoudun, quite close to both entrances to this zone. The entrance is a ship, that is docked in the starting area of the port.

If arriving via Observatory: Continue straight past the breach, go down the stairs to your left, cross the bridge to your left, turn right and it will be there, in front of 2 thespian enemies.

If arriving via Sohn District: Cross the bridge straight ahead of you, turn right and it's there, in front of 2 thespian enemies.

The first achievement you should get in Gehenna Prison is this:

Alpha
Defeated the Alpha


In theory this boss is missable, as you can simply run past him. However, it should be fairly obvious when you encounter him. He looks the same as some of enemies you encounter throughout the earlier parts of the level. However, you will drop down in to a small arena to fight this guy.

Next, after fighting through a rather fancy looking part of Gehenna Prison, you should encounter the main boss of this area, and receive the following achievement for defeating her:

Majesty
Defeated her Majesty


In the boss room, there is a mysterious locked door. If you look above this door, you will see an assortment of masks decorating the wall. Theses masks are also being worn by the fan wielding enemies you would have encountered on the way here. You need to kill these fan wielding enemies in the order they appear on the wall. Once killed, door will open, and you will get the following achievement.

Apologize you were born in my world
Defeat the seven deadly sins


If you're having trouble figuring out which order to kill these enemies in; at the time of writing this guide, there are 2 guides on YouTube for this achievement, by channels 'Gaming Tornedo' and 'The Real Vess'. I won't include either of their videos in my guide without permission. But you can search them up, and both guides are quite simple and easy to follow.

Through the boss room door is a space platforming section, where the EVA suit is recommended. However there is no achievements to be gotten here, just a cool sword.

The entrance to Dominion Baths is located in Arisen Dominion. Enter Arisen Dominion via the door from the Embassy. Go up the stairs in front of you until it plateaus, and you see a Arisen Congregator in front of you. Turn right and there will be a narrow hallway, with a Celestial Beast in it. The door is located in this hallway.

The first opportunity for an achievement you will come across is for hacking a turret. As far as I could tell, the first turret you encounter is not able to be hacked. However the 2nd and 3rd are. Basically there will be a small panel located behind the turret. You just need to run past them without destroying them, and then use the panel. This will disable the turret and give the following achievement:

Are you still there?
Hack a turret


In the room where the 3rd turret is (where you first encounter the black ops soldiers), there is a hole in the ground with platforms you can drop down to. Which will lead to an underground cavern. It's quite large and confusing. But basically what you're looking for is a pump room key. Once you open the door this key corresponds to. It will open a room with some infected soldiers and a large machine. There is a breakable wall in this room, that will eventually lead you to an optional boss. Again, this area is a bit confusing, but just keep in mind throughout this area that walls and stalagmites are often breakable, and you should eventually arrive at the boss, defeating it will earn you the following achievement:

Mama Slug
Defeated the mother slug


Continue on through the Dominion Baths area until you encounter the final boss. Killing them will get you our last achievement for this zone.

Mad Scientist
Defeated the Mad Scientist


After defeating the boss, there is an exit that will take you to the Mausoleum area of the Arisen Dominion. However, to continue with the DLC, you will want to use the rift in the boss room to teleport to the Embassy.

The core has multiple entrances. However the easiest one to access is located in the Embassy, right near the entrance to Arisen Dominion (you will have walked past it to access Dominion Baths). Opening this door requires credentials for The Core, which is found in Dominion Baths.

A large part of The Core zone is a fairly narrow cylindrical tower with many stories, that you enter and exit multiple times. The first time you exit this building, you will be in the middle of a staircase that leads both upwards and downwards. If you go up, there is rift that speaks to you, and asks you to complete a contract for killing people in multiplayer. If you proceed down the stairs, you can take your first left across a small bridge, then left again, through a door and jump up on to a ledge directly in front of you. There you can loot a white suit set. Equipping this suit gives the following achievement:

The one soul to collect myself
Find and equip the white suit


Continuing onwards and downwards, you should eventually encounter the final boss of the DLC - Fear. The mechanic with Fear, is that for each elder god that is still alive in the base game, his health doubles. Which means that if you are going for the achievement, and have all elder gods still alive, this will be a pretty long fight. I found that the Channeler of Light's pillar attack was very effective against this boss (especially when he is a blob in the center of the room). But most other methods I tried did minimal damage.

Killing this boss nets you the achievement:
Fear
Defeated Fear


and if all elder gods are still alive in your playthrough you should simultaneously unlock:
The only thing they fear
Defeated Fear at full power