One of the two NPCs outside Shadow Vault is called Gar. Quests you get from him is called Gar Achievements and they can only be completed in Shadow Vault. If you want to achieve this one as quickly as possible, just keep it in mind that submit quest of this floor and pick up the quest of next floor BEFORE you enter the next floor. For example, if you enter floor 17 without getting the quest, you won't get it even if you go back to Gar right now on floor 17, what means if you want to get the floor 17 quest you must pick up it when you are on floor 16.
What is this I don't even.