Add an action for building the docs

[ru-fu]

The docs are built on RTD, so we don't really require a separate check that ensures the docs build correctly - but not everybody has PR reviews enabled.

Also, our first check is currently the spelling check, which builds the docs before checking. With canonical/sphinx-docs-starter-pack#240, we can run the spelling check on existing output, so splitting the spelling check into building + checking the spelling of the existing output should not increase the build times, but make it clearer what is happening.