Adding automatic theme updates from a GitHub repository is actually pretty simple. The following function will hook into WordPress's native update system and grab the latest release from the repo of your choosing (if the version number has increased).
Place the following in your functions.php
// Automatic theme updates from the GitHub repository
add_filter('pre_set_site_transient_update_themes', 'automatic_GitHub_updates', 100, 1);
function automatic_GitHub_updates($data) {