Can Charge auto-update?

Plugin auto-updates are going to be integrated into Craft when the third party plugin store is released. Until then, you'll have to manually update Charge.

If you're running Craft 2.5+ you'll get notifications within the CP of any Charge updates, along with direct downloads of the updated version.

Because of the limitations currency on third-party plugins, you'll still need to actually unzip and move the plugin code to the appropriate place manually.