How do you pair the ring doorbell with Apple TV?

You don’t, at least not directly. If the doorbell is HomeKit compliant, you add it to the Home App on an iPhone or iPad and it should then appear in the HomeKit section of the Control Center on the Apple TV.

See here: link-> Set up your HomePod, HomePod mini, Apple TV, or iPad as a home hub - Apple Support

and here: link->Use tvOS Control Center on Apple TV - Apple Support