Good news on this one, you simply need to enter every room. You don't have to defeat any monsters that happen to be inside.
You'll want to do this achievement on either EASY or NORMAL, as I don't know if its even possible to win HARD or (especially) HARDCORE without skipping some tricky floors (e.g. those with Acrobat Mutants or Stretch Mutants) via the elevator. Obviously, EASY will be easier, as you'll get more keys through searchable loot as well as have more materials to craft keys.
As the above indicates, the only "difficult" part of this achievement is to get into all the rooms locked by keys. If you encounter one and don't have a key currently, try to remember what floor it was on so you can come back later with a key. Since keys take Gears and scrap metal to craft, try saving up a decent amount of scrap before the end of the game so you can craft any needed keys. As you'll have to visit every room anyway, this shouldn't be a problem.
If you get to floor 1 and this achievement didn't unlock, you missed a room somewhere. Try going back to floor 100 and walk down the stairs from 100 to 1 Since the stairwell should be clear at this point, you'll only need to worry about hunger when doing this check.
If you REALLY want this achievement on a particular run, there is an active perk called "Like They Do It in the Movies" which grants you a free door unlock every 100 turns. You really shouldn't need this perk, but its available if you're doing a run specifically for this achievement.
Final note: To put things into perspective, I was able to obtain this via my first complete NORMAL run without going out of my way for it, so you shouldn't have too much difficulty on this.