How to play Apple Music on AppleTV?

Provided you have an Apple TV 4 or later and once you’ve subscribed to Apple Music you need to log into your Apple TV using the same Apple ID as used for your Apple Music account in Settings>Accounts. You can then enable the service on your Apple TV in Settings>Apps>Music.

If you have a 4th Generation or 4K Apple Tv, you can access Apple Music directly in the Music App on the Apple TV as long as you are signed in to iCloud and iTunes & App Stores on the Apple TV with the same Apple ID that was used to initiate the Apple Music subscription service. :

https://www.imore.com/how-use-apple-music-music-app-apple-tv

If you have an earlier model of Apple TV, then it does not directly support Apple Music, but you can Airplay Apple Music tracks from a computer or iOS device that has them.

https://www.imore.com/how-airplay-music-apple-tv