If you have beaten the Wailing Critters dungeon before and your character is at least level 110, Breanni in Legions Dalaran will offer you a breadcrumb quest to fly to the Deadmines in Westfall.
There, Marcus "Bagman" Brown
will greet you and send you to the actual dungeon.
The dungeon's 9 fights are much more diverse in comparison to the Wailing Critters, but are overall very comparable in difficulty. If you have a large roster of pets, you won't have any trouble beating it.
The unlocked challenge mode works exactly like before - no healing or reviving pets within the dungeon. Completing it however will grant you one Old Bottle Cap
, a new currency that you can directly turn it at Marcus for three new pets:
To get all three (non-tradeable!) pets, you will therefore have to run the dungeon 7 times. Once on normal mode and then 6 challenge modes.
Good luck and enjoy the new dungeon!