Update (November 2023): an updated guide on installing Google Play Store and apps from Play Store on Amazon Fire tablet
Please check this updated guide on how to install the Google Play Store on a Fire tablet.
Update (July 2019): Amazon Prime Video app now supports Chromecast
As explained in this guide, the Amazon Prime Video app on Android and IOS now supports Chromecast.
However, the Prime Video app on Fire tablets still does NOT support Chromecast. And you cannot install the Google Play Store version of the Prime Video app on a Fire tablet.
Update (2016/02): Android screen mirroring works on Fire tablet 5th generation and later