After 20 attempts and 5 complete screen clears, I knew this achievement was tricky. This is how I finally unlocked it on Level 1:
1. Kill the Fryga (the green dragon).
2. Kill one Pooka (orange, round thing). Leave two Pooka's alive.
3. Set off the rocks to fall to get rid of them. That way they break, freeing up space for you to clear later. Again, free the rocks so that they are wasted and break and won't kill anything. Do NOT crush monsters with them.
4. As you dig dirt, if the two remaining Pookas are closing in on you, pump them up a little bit to slow them down. Three pumps is the max. No more than three or they will explode. If you kill all but one monster, it will run away and the level will end, resetting all the dirt you worked so hard to dig. Keeping two Pookas alive will ensure that the level lasts long enough for you to dig completely.
5. Retrace the entrance shaft.
6. Retrace Dig Dug's starting point.
7. Walk the surface.
8. Retrace the monster rooms.
9. Clear the dirt.
10. Kill the two remaining monsters.
11. Finish and move on to Level 2. This is VERY important. If you clear the dirt in Level 1 and commit suicide before reaching Level 2, you can kiss that achievement goodbye.
12. At Level 2, feel free to commit suicide. When you get the "Game Over" message and go to the achievements screen, the achievement will pop.