What the achievement asks
Complete the Black Empire campaign storyline, which ends with Ny'alotha, the Waking City: The Corruptor's End. Ten points, both factions.
This one is not tied to your account. The character that finishes the campaign is the one that gets the credit.
The legendary cloak is the whole difficulty
The final step sends you into Ny'alotha to kill N'Zoth, and the way in is offered by an NPC rather than the ordinary raid finder.
If the legendary cloak from the patch 8.3 questline is not in your bags, that option is not offered. No message explains why: the choice simply is not on the list.
The cloak has to be in your inventory. It does not need to be equipped, and having it in the bank does not work. Take it out, talk to the NPC again, and the option appears.
This catches almost everyone coming back for the Battle for Azeroth meta years later, because the cloak stopped being useful the moment the expansion ended and most people filed it away.
The same trap, one achievement over
The Waking Dream, the raid achievement for the last two bosses of Ny'alotha, is blocked by exactly the same thing for exactly the same reason.
The two are usually finished in the same sitting, so the sensible order is: find the cloak, put it in your bags, then go and do both.
What has to be done first
The Black Empire campaign is the patch 8.3 storyline, running through the assaults on Uldum and the Vale of Eternal Blossoms. It has to be worked through before its final chapter exists, and that campaign is what awards the cloak in the first place.