I recently started working on porting Doom to the Apple TV after I realized that no one had released any source code so someone could effectively side load it on their Apple device. I also have game controller support working.
As you can see the game is definitely running and works great with an mFI game controller.
Once I have cleaned up the menu system and crashes I will post a link to the GitHub repo on my site at http://mobilefission.io with instructions. Ultimately, as long as you have a Mac, you will be able to use Xcode to build it by simply adding the WAD file from a copy of Doom you have already purchased.