How to host your WordPress theme on github and make my theme updatable for my client

You can use WP Pushar to update the theme from github. It will give you synchronize button in the dashboard to synchronize it with github repo.

Here is overview video you can use WP Pusher with your site.

Here is article that explaining how you can update theme from github using WP Pusher. Also watch that video to understand it more clearly. Video