Really happy to see that GitHub ship more often lately. This feature seems really well done - I'm a bit bothered that the API is reserved only for apps installed through their marketplace - it'll be impossible for instance to report a custom Jenkins job there.
The Checks API is available for all GitHub Apps (the next-generation of GitHub integrations, see: https://developer.github.com/apps/), not just apps on the Marketplace. That said, I'm not aware of a Jenkins plugin that acts as a GitHub App (yet).