What the achievement asks
Be expunged from the City of Threads by the guards. Ten points.
You have to get yourself thrown out, and the city will not oblige while you smell like it does.
The reliable method
Everyone moving safely through the City of Threads carries a pheromone disguise. The guards ignore you while you have it.
Dismount first. Standing next to a guard while mounted used to strip the buff and no longer does. On foot, stand close to a guard carrying the true sight icon above their head and your pheromone buff is removed for a minute.
Then linger. A Sureki Eradicator finds you and begins casting Eradicator's Mark. At ten stacks you are thrown out and the achievement fires.
The Eradicators are not spread evenly through the city. Players place them in the upper north-eastern part of the map, so that is where to go and wait rather than wandering the whole place with no buff.
The accidental method
There is a second way, and plenty of people got it without meaning to. During the Azj-Kahet campaign quest Like a Spider on the Wall, at the step where you attempt to meet Meesk, you talk to Nana Lek'kel.
Pick the rude conversation options rather than the ones flagged as advancing the quest. She loses patience, webs you, and has you expunged, achievement included.
Getting back on your feet
Being expunged leaves you with a Trapped debuff. Click the webs on your character a few times to break out of it. If you would rather not, it falls off on its own after about five minutes.
Where it fits
One of the pieces of You Xal Not Pass, which rewards the Shadow of Doubt mount and is part of Worldsoul-Searching, the War Within meta.