What the achievement asks
Defeat Umbrelskul in The Azure Vault, on Mythic or Mythic Keystone difficulty. Both factions.
A keystone run counts at any key level, and it counts whether or not you beat the timer. Reaching the last boss and killing it is the whole of it.
The one thing that catches people out
This is not account-wide. It has to be earned by the character that runs it.
That is the opposite of how Shadowlands worked, where the dungeon achievements counted across your whole account and only Tazavesh did not. Here all eight Dragonflight dungeons are per character.
The practical consequence is worth stating plainly: spreading the eight dungeons across your alts leaves you with eight unfinished achievements and no progress at all on the meta. Pick one character and run all eight on that one.
Mythic is not in the automatic queue
The Dungeon Finder offers Normal and Heroic. It does not offer Mythic, so no amount of queueing will get you there.
You need a premade group instead: open the Group Finder, go to Premade Groups, choose Dungeons, and either list your own group or join one. Alternatively, any Mythic Keystone run counts, so if somebody in your guild has a key for The Azure Vault, going along on that finishes this just as well.
What it takes today
At current levels a Dragonflight Mythic dungeon is a formality rather than a challenge. There is no meaningful gear requirement any more, the base Mythic difficulty sits well below what a modern character brings to it, and a group usually clears it faster than it took to form.
Groups for these form constantly, because everybody working through the meta needs the same eight dungeons on the same character. Listing your own group is often quicker than waiting to be accepted into someone else's.
Where it fits
One of the eight under Myths of the Dragonflight Dungeons, itself a piece of A World Awoken and the mount Taivan. The other seven are exactly the same job in a different dungeon, so plan them as one evening rather than eight errands.