Apple TV Video Compression Process
It is up to you whether you want to export from Final Cut Pro as a QuickTime file and then open that QuickTime file in Compressor, or just Send to Compressor directly from Final Cut Pro. If this is just a freestanding video that you have downloaded or captured then you are just going to want to open that video file in Apple Compressor right off the bat.
Go down to the Settings pane and select the Apple folder. Go to Apple Devices, which will be the top option above DVD. In here the top option will be H.264 for Apple TV, which is right above H.264 for iPod Video and iPhone 320 x 240 (QVCA). This format is a standard H.264 that will be specified for Apple TV, and it is listed in its details as "H.264 1280 x 720 Video @ 5Mbps, progressive, Audio is 44.1 kHz, stereo." This indicates that all settings that should be optimum for Apple TV will be included in this video codec. Drag and Drop H.264 for Apple TV up into the project pane above where it says "Drag Settings and Destinations Here."
Make sure to go up to Target and select Destination, which will then take you to a menu where you can select destinations in your computer or attached digital storage devices. This is the destination where your compressed video will go when Compressor has completed it. Just hit the Submit button in that window and the video compression process will begin, and it will spit out a single file in the same way that a QuickTime compression would for upload to YouTube or Vimeo.
From here it is up to you how you want to employ the file onto Apple TV, but you can likely just transfer it onto a digital storage medium and then plug it into your Apple TV.