Progress through the level normally and skip picking up any of the clues. Certain doors in the level will set the clock forward 10 minutes, and checking just about every room in the level will insure it completes. When you get to the "final room" (the one showing the anatomy of Hat Kid) just turn your tookis around and leave, 10 minutes should advance.
Big Note: When you enter the room where the voice on the phone tells you to save the dog, don't do it. Just leave. This room 100% works against progress towards this particular achievement, and despite what the guy tells you, time does not advance. Saving the dog and talking to it afterwards will automatically get you a clue, so don't bother.