There hasn’t yet been an official version of the Amazon Instant Video app released on Google Play for Android TV devices like the Nexus Player and Razer Forge TV to install. However, there was an official version of the Amazon Instant Video app installed on Sony TVs that have the Android TV software installed on them.
User brar.arsh on the XDA Developers Nexus Player forum took the APK for the Amazon Instant Video app from his Sony TV and uploaded it so other Android TV users can install it on their devices. Users have confirmed it works on the Nexus Player and the precursor to the Nexus Player, the ADT-1. The APK should work on the Razer Forge TV as well, assuming you don’t have any other Amazon app installed on the device.
To install the APK you can use either Air-Share & Air-Launch or install ES File Explorer on your Android TV device. From the program of your choice, launch the APK file to start the installation process of Amazon Instant Video.