During Mission 5: Clean House, upon going from the 1st floor to 2nd floor (after killing 6 total enemies), before moving forward, you'll want to stop halfway up the stairs and shoot out the door in the hallway above revealing an enemy inside. Kill this enemy. Additionally, throw 1-2 grenades in this room for safe measure to take out additional units. With any enemies near the door cleared out, you are now free to move forward and Alpha 3-2 should make it past without being injured.
Clear the bedroom as well as the corner of the room before your character announces that the floor is clear, and proceed to the third floor to unlock the achievement.
Possible on any difficulty, but recruit is probably recommended to make it as easy as possible.