What the achievement asks
Complete each of the four covenant assaults in the Maw once. Ten points, account-wide, both factions.
This is the least demanding piece of On the Offensive and the one to start with, because it needs nothing except turning up.
How the assaults work
Two assaults run each week, one starting Tuesday and one Friday, and each lasts about three and a half days. Four covenants take turns, so any given covenant's assault comes round once a fortnight.
Each assault offers four quests, drawn at random from a pool of around ten. That second layer of chance is what made this branch notorious: the quest you need may simply not appear for several of its own assaults in a row.
What that means for you
Two visits a week, and a fortnight to see all four covenants come round. There is nothing to hunt, no quest to hope for, and no way to speed it up.
An assault is a short chain of four quests in one corner of the Maw. Do them, and the covenant is ticked off.
Do it alongside everything else
Since you have to be in the Maw twice a week anyway, this is the achievement to run in the background while you collect the others.
The two chest hunts, Up For Grabs during the Necrolord assault and Jailer's Personal Stash during the Night Fae one, both happen inside assaults you have to attend for this. Doing all three together gets you three of the four pieces of On the Offensive with no reliance on luck at all.
The Venthyr assault has been reported failing to complete, with the mirrors at the end not spawning to let you reach the final boss. If it happens to you, it is not something you did: come back on its next rotation.