tvOS betas are more difficult to install than beta updates for iOS and OS X. Installing the tvOS beta requires the Apple TV to be connected to a computer with a USB-C to USB-A cable, with the software downloaded and installed via iTunes or Apple Configurator. Once a beta profile has been installed on the device through iTunes, new beta releases will be available over the air.
As a minor 9.x.x update, tvOS 9.2.2 appears to focus on bug fixes and performance improvements, much like tvOS 9.2.1, the current version of the tvOS operating system. According to Apple's release notes, the first beta included bug fixes and security improvements and no outward-facing changes were spotted.
We'll update this post if any new features are found in tvOS 9.2.2 beta 2.