Here is the full list of all 12 Death Stair achievements.
Lose 1 Online Match
Play a 4 Player Online Match
See Every Award Once
Win 5 Consecutive Party Mode Matches
Play 200 Matches of Any Type
Play a Local Match With 4 Players
Get Hit as a Runner 5000 Times
Use 50,000 Ammo Across All Modes
Get Hit as the Runner With Every Ammo Type in the Game
Invite a Friend to an Online Match
Play as Every Character
Play Death Stair For Over 2 Hours