How do I authorize my new tablet after deauthorizing all my other computers?

You only authorize computers on which the tablet would be used.

Authorization

iTunes Store- Authorize or deauthorize your Mac or PC.

In iTunes you use the Authorize This Computer or De-authorize This Computer option under the Store menu in iTunes' menubar. For Windows use the ALT-S keys to access it. Or turn on Windows 7 and 8 iTunes menus: iTunes- Turning on iTunes menus in Windows 8 and 7.

Authorize your computer in iTunes - Apple Support

You don't authorize iOS devices, only computers. Is there a particular issue or question you have regarding your device?

Cheers,

GB