Today, a Reddit user that goes by the name of /r/TheUnkn0wn0ne posted a link to the XDA forums revealing that an unofficial CyanogenMod 12 ROM is now available for the Amazon Kindle Fire [1st edition]. The build is only compatible with the Kindle Fire when setup as OtterX.
The build, which is still in Alpha stages does function quite well, but there are known issues you may want to read before diving into Android 5.0 Lollipop on your Kindle Fire. Some of the known issues are Google Play Movies not functioning, device boots up in “landscape” (which may change), transparency settings are disabled due to low RAM of the device (being worked on), and high battery drain when the device is set to idle. There are reports of 12%+ draining per hour.
Be careful if you want to do this to your Kindle Fire, because once you have your device on OtterX, it is very hard to flash it back to the stock ROM. Some images of the Kindle Fire running CyanogenMod 12 can be seen below.