Book of Demons

This is possible to do very early on, especially for the archer who can clear out monsters way ahead. Finding the short path from the beginning to the exit without backtracking is luck of the draw. But as soon as you find the exit, you can leave the level, getting the first achievement. Immediately go back to the beginning of that level and just walk from the beginning to the end 4 more times to get the other 2 achievements.

This is most easily done in the end game in a very short dungeon in hell. Many hell dungeons are laid out purely linearly, so it's pretty easy to go through them without backtracing your steps at all. The charge ability can make this a bit easier to accomplish.