how do i hook up amazon prime to apple tv?

You can't directly as they have not made an app for the 4th gen. You would need to use airplay from an IOS device or mirror your computer

How to AirPlay content from your iPhone, iPad, or iPod touch - Apple Support

Use AirPlay to display video from your Mac on an HDTV - Apple Support

Long time coming but Amazon Prime Video is now available on Apple TV under the “Prime Video” app that launched on December 6 2017.

Apple's set-top box is one of the last platforms to support the Amazon Prime Video app, which is now available on third, fourth, and fifth-generation Apple TV models.

Movies and TV shows in Amazon Prime Video can be added to your Watchlist, which is synced across all of your devices, and the Apple TV's Siri Remote can be used to search for Amazon content from within the app.

Universal Search is also supported, so Amazon Prime Video content will come up in systemwide searches on the Apple TV, and integration with Apple's dedicated TV app is included in countries where the TV app is supported, with access to Up Next syncing and content recommendations.

Integration with the TV app is nice, as is Universal Search, because both features let you access Amazon content without needing to use the Amazon Prime Video interface.

For the 4K Apple TV and supported television sets, supported Amazon Prime Video content plays in 4K HDR.

I have a 3rd Gen Apple TV set to update automatically, and the app appeared automatically.