Does the Apple Thunderbolt Display charge the new MacBook Pro 2016 over this adapter or should I use an external power supply for the computer?

No. The thunderbolt display does not provide power back over it’s thunderbolt connection.

The thunderbolt display can only provide power via its Magsafe connector, but that can’t be used to provide power to the USB-C ports on the MacBook Pro’s. (unless some sort of MagSafe to USB-C adapter comes along, but I have not heard of such thing yet.)