Unlike the Pacifist and Silent Hunter achievements, you don't have to do this right away.
You have to eat nothing but animals that you've killed or caught for 25 days in a row.
DO NOT eat any raw fish or meat you find in fridges this will reset the clock, Including fish or plants that you find while beachcombing.
Eating or drinking any processed foods is a big no no for this achievement, including coffee and tea.
Consuming any medical item is fine, as long as it doesn't give you calories (drinking rose hip, bark, and reishi tea is fine as long as you've made it yourself).
It's best to get yourself established on the island first and have sort of firearm to make hunting easier, if you haven't found one then fishing is suitable alternative.
This is much easier to achieve late game as you would either have eaten most of the processed food or it has gone off by the time you've found any, and you would be relying on animals to keep you alive.