If you've passed part or all of the game.
You should wait until the Donkey will deliver a fresh corpse. When you’ll come to the corpse hatch, you’ll find not one, but two bodies. That's the beginning and first quest of the Better Save Soul DLC.
In the beginning.
After the beginning of the game and passing the tutorial, you should have an open Church (upgrade your graveyard to 5 points and talk to the Bishop), and repaired corpse hatch (inside and outside). After that, when the Donkey will bring you a fresh corpse, you’ll find not one, but two bodies. That's the beginning and first quest of the Better Save Soul DLC.