Update (2016/02): Android screen mirroring works on Fire tablet 5th generation
For Fire tablet 5th generation ONLY (you can check yours in Settings– Device options–Device model), you can also use screen cast for Chromecast (Android screen mirroring). You can run Play store and update Play services. Please follow this guide to run one ADB command.
For all previous generations of Kindle Fire, Kindle Fire HD and Kindle Fire HDX tablets, you cannot run Google Play store app. Screencast does not work either. If Chromecast does not work after install the 4 APKs, you can try the latest Chromecast APK and Google Play service APK (both are from APKMirror.) This Chromecast app APK matches this Play services APK.