The new tvOS 12.1.1 developer beta can be downloaded onto the Apple TV via a profile that's installed using Xcode. Subsequent betas can be downloaded over-the-air.
We're not yet sure what fixes and changes the tvOS 12.1.1 update might bring, but it's likely to focus on bugs that were not able to be addressed in the tvOS 12.1 update.
Apple provides little information on its tvOS software and tvOS updates have historically been minor in scale, so we may not know what's included in this update. Nothing new was discovered in the first two betas, but we'll update this post if any new features are found in the third beta.
Update: Apple has made a new public beta of tvOS 12.1.1 available to its public beta testers.