To unlock Broom Closet, you must complete the coop map Soleptor Excavation. You must have an EGO rating of 400 to unlock this coop mission, as well as progressing past a certain story quest within ACT2.
To access coop maps, from the in game menu press right on your dpad. Select matchmaking, then coop maps, then Soleptor Excavation..
Then, play through the mission. This is an easy coop mission, with even the boss a bit of a breeze. Once the boss is dead, you will receive your achievement.