The API is documented in the ACME draft: https://tools.ietf.org/html/draft-ietf-acme-acme-09 . There are newer drafts but I believe that 09 best reflects the current state of the v2 API (correct me if I’m wrong).
There are some divergences to the draft in the actual Let’s Encrypt API, which are documented here: https://github.com/letsencrypt/boulder/blob/master/docs/acme-divergences.md
Between those two and referring to the code of (well-written) existing clients, you can piece together how to build a client.