You will need three things to add Version Checker support to your mod: a uniquely-named .version file, an entry in data/config/version/version_files.csv pointing to it, and a master .version file somewhere online to check against.
The CSV is required due to how Starsector handles file loading and is trivial to add. Just create the file
data/config/version/version_files.csv in your mod folder, then open it in a text editor and paste the following (make sure to change the .version filename to whatever you're planning on using):
version file
insertmodnamehere.version
Then create said version file in the root of your mod directory. You can find an example version file with a description of the necessary fields
here.
Once you've finished, upload this version file to the internet somewhere and change "masterVersionFile" in your local copy to point at its URL. The only part that is read from the master is the mod version data, so don't worry if the masterVersionFile URL in the online file doesn't point to itself. Whenever you update your mod, just re-upload your local .version file with the new version data and all users running Version Checker alongside your mod will be notified of the release when they next start Starsector.
That's it, you're done. Now all you have to do is remember to keep these files up to date. It helps to add a comment next to the "version" line in mod_info.json reminding you to change both files when you update.