Lapin

You can cheese the deathless runs by exiting and re-entering your save file after you die. As long as you don't die (or pause) before reaching the next screen transition on your successful attempt, you will still get the achievement. If you die or accidentally pause, quit and re-enter.

This also means you can get the 'Pancake' achievement on the same run as 'Sympathy for Mr. Vengeance'.