I know this topic has been brought up many times but I think it could still be improved.
Checking the readme file for the compatibility info is fast, but it isn't always up to date. Plugin authors do not need to update the plugin version to update the compatibility info in the readme file, so the readme is often outdated.
I understand that the plugin API may be getting it's data from the SVN trunk, which authors don't always update either, but maybe a combination would work. Checking the API is also much slower, so I have an idea for that too.
First, find out what the latest version of WordPress is. Then check the readme file for version information. If the readme file says it is not tested with the latest version, then check the API to see if it's been tested with a higher version - if so then use that version instead of the readme file version.
Since calling the API takes time, I also suggest caching the API result in a database transient. Make it expire weekly, but also provide an option to clear out the cache.
https://wordpress.org/plugins/better-plugin-compatibility-control/