Please refer to this thread:
Download this apk: [apk, updated on Jan-17-2016, now version 1.4.0, it will expire on Feb-29-2016]
Instructions: (good for fire TV box and stick)
(1) Enable ADB and USB Debugging - go to Amazon TV, Settings, System, Developer Options > tun on ADB Debugging and USB Debugging
(2) Now, you need to find the ip address of the FireTV device, System > About
(3) Now from another PC, you can use adb command to install the YouMap Chromecast apk.
adb connect your-ip-address-of-firetv
adb install YouMapCastReceiver.apk
(4) reboot fire TV, and it's ready for casting.
Have fun !