Since version 1.6 the plugin can be updated through the administration pages, just like the free plugins from WordPress.org. You will be notified about new versions and can install them by clicking on the update link. Earlier version need to be re-downloaded from the marketplace where you bought Tag Groups Premium.
In our tests, the update works smoothly even on multisite installations. In the case that you should experience problems, try to deactivate and remove Tag Groups Premium. Then install the new version as zip file, that you have downloaded from the marketplace.