What the achievement asks
Unlock every door in a single Zskera Vault within one week. Ten points, account-wide.
A vault has 28 or 29 doors across its four floors, so this is 28 or 29 Zskera Vault Keys spent in the same vault, in the same week.
Bring the keys with you
The mistake is walking in with six keys and hoping. Farm or buy the full thirty before you enter, and the achievement is a twenty-minute visit instead of a week of trips that ends with two doors unopened when the reset wipes your progress.
Keys are account-bound, so anything an alt picks up can be posted over. And since patch 10.1.5 the renown envoys at Dragonscale Basecamp sell them for 50 Dragon Isles Supplies each, which puts a full vault at about 1,500 supplies. That is small change for anyone who has played the expansion at all, and it removes the only real obstacle here.
Keys otherwise drop from the Forbidden Reach's rares, near reliably, and from its treasures.
You do not have to solve anything
Only the unlocking counts. What is behind each door, whether you loot it, and whether you work out its puzzle make no difference to this achievement.
That said, open the rooms as you go rather than sprinting: several of the books for Librarian of the Reach and While We Were Sleeping only spawn inside vault rooms, and you are already standing in the one place they appear.
If it does not pop
Open every door and get nothing? Leave the vault and go back in. The achievement lands on re-entry. It is a display quirk rather than a lost run, and there is no need to spend more keys.
Where it fits
One of eleven under You Know How to Reach Me, a piece of A World Awoken. The same run counts towards Door Buster, which wants 150 doors in total, so clear a full vault every week rather than a partial one and the two finish together.