To get this achievement, it requires you to open the timed doors in a certain amount of time before they shut up shop. I would recommend rushing for this and obtaining it in the second stage (Promenade of the Condemned or Toxic Sewers) before 2 minutes. The times are always static.
Promenade of the Condemned: 2 minutes
Toxic Sewers: 2 minutes
Ramparts: 8 minutes
Ossuary: 8 minutes
Black Bridge: 15 minutes
Stilt Village: 17 minutes
Graveyard: 19 minutes 30 seconds
Forgotten Sepulcher: 26 minutes
I've added a video for visual guidance and some helpful tips on how to obtain this achievement: