I'm pretty sure this doesn't have to be done in one run as the achievement popped for me when i accidently left the dungeon by walking into a ladder. On that run i had only been to two corners but had been to the others on a previous run.
The first "Deadly Dungeon" you come across is in the deep dark wild after round 4. It's a blue portal (doorway). If you hug the left wall as you're moving to the green objective you will come across it about halfway through the zone.
EDIT: you have to leave the dungeon for the achievement to unlock