Yes! You can watch Fullscreen videos on your AppleTV by downloading the app or by using AirPlay.
To download the app, you must be on a 4th generation (or higher) Apple TV. You can find Fullscreen in the Apple TV App Store. Download the app, then simply log in to your preexisting Fullscreen account, or sign up for a new Fullscreen account.
For Airplay, make sure your iOS device and AppleTV are on the same Wi-Fi network. Then when you start playing a video, tap the Airplay icon in the bottom right hand corner and select the AppleTV you want to stream to. Use the controls on your phone to play/pause and fast forward/rewind.