The Dungeon is accessible once you have proceeded with Snake’s quest line a bit. After meeting him the first time, you’ll need to come back with 5 Faith to convince him you’re the Keeper. After that, you’ll need to get him to help you open the door he’s after, which will involve some work with the Astrologer and a key. You’ll also need to have upgraded your Graveyard enough to open the Church, since you need to use the study desk in the Church basement to study the key. Once you‘ve done that, you follow Snake past the door and to the Sacrifice Zone. While he’s checking out what he’s interested in, you can finally obtain the Keeper’s diary. Once that’s obtained, the Dungeon is finally available to you, once you speak to Snake again.
From here on out, it’s easy going. Just explore each level of the dungeon for all monsters, kill them, and find the door to the next level. Once you reach level 5, you’ll find a chest in front of the door to the next level. Killing all enemies on dungeon level 5 and opening the chest will award you this achievement.