You can use Chromecast to view Apple TV, but this particular combination of streaming service and streaming device is a little tricky. Chromecast and Chromecast Ultra devices are intended to receive video casts from a phone or computer, but the Apple TV app is intended to cast only via AirPlay, not Chromecast.
To view apple tv Chromecast, cast from the Chrome browser rather than the Apple TV app. The Chrome browser includes the ability to cast compatible webpages to Chromecast devices, and the Apple TV web player is one of them.