Using this seems easiest:
https://github.com/actions/toolkit/blob/master/docs/problem-matchers.md

This looks like quite a good example to base it on:
https://github.com/TrueBrain/actions-flake8

There are a couple of Codespell actions already:
https://github.com/marketplace?type=actions&query=codespell

Both by @plettich .

One just uses the dictionary, and so presumably would need an update for out multi-dictionary, won't have the same options etc (as well as only creating review comments):
https://github.com/marketplace/actions/codespell-action

The other one uses Review Dog, which seems like an unnecessary overhead now the problem matchers exist:
https://github.com/marketplace/actions/run-codespell-with-reviewdog