I felt a guide for the achievement needed a bit more detail so here goes.
First a few pieces of background information. I started a fresh save file, and I have only played the game since the patches to repair the broken achievements have been active. This was my 4th save file, (I accidentally over wrote my Elder save file I was using to get the Deer Boss achievement so I deleted them all and started fresh).
Solution: I ran to the right from the start, learning my first skill straight away as usual. I then proceeded to kill every good creature, (confirmed by a red 'flash' when they die), until I reached the first human you can talk to,
, this gives you a nice respawn point.
At this stage my status was Immoral, press
to check, the red bar halfway from the centre to the left antler, (it only looks like a horn at this early stage). I then let the next bad creature kill me, came back as a beaver at our respawn point. Got to the first bad creature and let them kill me again, this time coming back as a fawn. Kept this process at the first bad creature going until the achievement popped as I became a beaver for the 3rd time.
Total time was about 5 minutes. In a previous save I had definitely been reincarnated as a beaver, though I cannot categorically state why this was not counted, it could be because I had deleted the save...or that it has to be on the same playthrough. Also I did not collect the extra life to the left at the start as I assumed it would ruin the reincarnation, but it may not!
Please message with any improvements.