While this achievement is not like the others and only requires you to beat it (with no restriction on deaths or time), it will still be a tricky achievement for some. My first run of Bowhemoth took an hour and 45 minutes with 413 deaths.
Advice: Make sure you have plenty of time to complete this in one sitting. There are no saves throughout the mode. If you die in a room you will reset to the beginning of the room (not the whole Bowhemoth run), but if you quit in the middle, you cannot pick back up in that room.
Take your time all you'll eventually get the hang of it!