If your existing installation is relatively uncustomized, then I suggest you plan to do your upgrade by creating a new test installation, from github, making a backup of your existing database, and then upgrading that database using the github installation. If everything is good, then the new github installation becomes your live site. If you have any old customizations, you can port them to the github installation and perhaps commit them as local commits so you can track them in the future.
It’s basically the same idea as generally upgrading from source tarballs; you’d still want a test installation anyway, and backups. You can’t really “update” a non-repository installation to a github installation directly.
Cheers,
Jason