As I previously warned, and will warn again, installing a beta operating system is for developers who are testing applications. It is not for individuals. There will be bugs and issues, sometimes rendering your device near useless.
You will need to be part of the iOS Developer Program with Apple and have access to the proper files, as well as access to registering the UDID (Unique Device Identifier ) of your Apple TV.
There is an installation guide posted in the developer portal that instructs you to use iTunes to load iOS 6 onto your Apple TV, but I've ran into nothing but issues using that method, so I'm going to show you how to install iOS 6 via Xcode on your Apple TV. In fact, you can use this method to install iOS 6 on any test device, should you run into issues when attempting to install using iTunes.
Launch Xcode, then open the Organizer (in the menu bar click on Window > Organizer).
Next, you'll need a micro-USB cable to connect the Apple TV to your Mac. With the HDMI cable disconnected from the device, turn it on and wait for the white light to stop blinking. After the light is solid, plug the micro-USB cable into the port just below the HDMI port.
Make sure you have the Devices tab selected in Organizer. Your Apple TV will appear on the left-hand side of Organizer. Click on the device name. You'll need to register the UDID with Apple before continuing. The quickest way to register it is by clicking on the Use for Development button in Organizer.
After registering the UDID of your Apple TV, the software box will become live. Click on it and select Other Version.
Select the IPSW for the respective version of the Apple TV you are loading iOS 6 onto.
The loading process will take some time. The most important thing to do is wait; don't get impatient and unplug your device too early. Once the new OS is loaded, you'll see the above message below the OS version. Don't worry. You can't activate it through Xcode, or iTunes for that matter, you have to reconnect your Apple TV to your TV and go through the entire setup process; you won't be able to restore it from a backup.
If you run into issues, visit the Apple Developer Forums for some troubleshooting help.