Right after getting the Dead Eye achievement you'll turn around and be in the Void. To the left is a dead cultist with a note to read and a key. Getting the key makes a waypoint to his locked room appear. Nearby is a cliff looking off into the distance with a telescope on it; look through the telescope to get a marker that will lead you to the Outsider.
Go to the dead guy's room, and read the notes and play the audiograph. You'll see the Outsider's mark written on the board, as well as lots of symbols with numbers written next to them. Three of the symbols make up the Outsider's mark. Looking at the symbols from left to right, those are the combination to the safe in the room.
Make your way to the waypoint taking you to the Outsider. You'll end up deeper in the Void now. Once inside, follow the waypoint to the Outsider. Just before him, you'll encounter
Note: Make a save here so you can reload and get Deicide if you don't already have it unlocked.
Now go up to the Outsider and interact with him. You'll be given two options: to kill him (the first choice; unlocks Deicide) and to spare him (the second choice). Choose the second option.
You'll now be prompted to
You'll be given a series of dialog choices. Choose the first each time. Eventually,
Sadly, we don't get to hear it. At this point, the end scene will trigger and the achievement will pop.