INTRO
There are two achievements related to MoM - "What Final Horror" and "R'lyeh Nice to Meet You". They also seem to be quite elusive, with less than 2% of players having them at the time of my writing this. My method for getting them is a bit time-consuming, but practically foolproof. You will have to do it twice to get both achievements, as starting the airplane will take you to the end credits.



THE METHOD
To prepare, you will want to follow this sequence of events:

1. Enter Dagon, get the Compass, and explore the levels
Hoard as many artifacts and bullets as you can. Use any flowing fountains you find, and most importantly, get the Materialize spell. If you don't find it in Dagon, try Halloween. If you still didn't find it then it may be a good idea to just suicide and try again.
Dagon does not take much time to explore and beat, so get everything in this world and then exit through the book on the pyramid altar.



2. Enter the Halloween world and explore it
Again, grab all the fountains, artifacts, keys, and bullets you can. You can also grab a pickaxe/dynamite here. I recommend a Flintlock Pistol as well.



3. Assemble your loadout
You want the Compass, Med Kit (optional), and the Climbing Boots. I recommend not deviating from this setup. If you can't find the Med Kit, the Tinning or Locksmith Kit can also help, but I recommend having the Med Kit. You will also want to have at least 75 bullets, 15 keys and 150 artifacts. If you have had lots of failed attempts at MoM, pull some extra artifacts out of your Bank. Feel free to farm Dagon/Halloween for a while to give yourself a surplus of items.
Once you're sure you're well-prepared, enter the MoM book.



4. Enter MoM and forge a path
Once inside MoM, mark the entrance by Materializing a block. These blocks will never despawn. Get in the habit of marking each entrance and exit. You'll also want to place at least one block per screen to ensure a clear and visible path between entrances and exits. If your route takes you through a pit, mark the rim of the pit, to make it obvious that you need to ascend/descend through it.

Pay attention to where the Compass points. You want to dig/blast the most direct route possible between each level's entrance and exit. Since you have Materialize, you can build big staircases to make a direct path, as well as walls to block enemies from getting to you. Take it slowly and stealthily, and make sure you're going the right way before you place a Materialize block, or else it might confuse you later. Whenever you find Dynamite or a Pickaxe on the ground, pick it up just for long enough to continue your path. Or, juggle-drop them down toward the level exit, taking care that they don't bounce off walls and hit you. Then, take them into the next level just long enough to use them to create the next part of your path.

Since you farmed items before you entered MoM, leave all the enemies dead after killing them. This strategy coupled with the Climbing Boots allows you to explore the upper levels at your leisure, and helps prevent you getting screwed at the end of your run.

Hit the wooden locked doors 8 times with a dagger or hatchet to break them. (Iron locked doors still need a key.)

When you start seeing Shoggoths, prepare some safe havens using Materialize blocks. These can be nothing but 2-block high platforms, or you can enclose them to protect yourself against ranged attacks as well.



5. Get the Keys
Once you're at the final world, lurk behind cover and kill everything around the altar before approaching it.

If you're trying to get "R'lyeh Nice to Meet You", backtrack to the first level and grab the Soapstone. If you found any shops selling the Destruction Amulet, buy that too. Using the Amulet, rather than a pickaxe or dynamite, helps de-randomize this part. Take the Amulet back to the final level and use it to find the secret room, then go back to where you dropped the Soapstone and take it. Now, use the book to return to the Library, enter Cthulhu, and talk to the first shopkeeper you see for the achievement.
Otherwise, grab the keys and start following your breadcrumb trail of Materialize blocks back to the surface. Azathoth will obscure your vision, but unlike the Azathoth in the Asylum, this one cannot hurt you. I lost a few good runs by not knowing this. Hide in your Safe Havens when necessary, and lure Azathoth to them so he isn't blocking your vision while you're finding your way.

Once you have the keys, the enemies on the upper levels will respawn, so you will have to fight many of them again. Use your safe havens, build walls, and climb back up using the stairs you created on the way down. As before, don't loot the dead enemies. Since you prepared extensively, this part will be a piece of cake.



6. Win!

Once you're at the surface, mouseover the airplane and hit Use. That's it!
I hope this helps someone. :D