I just finished installing the CyanogenMod ROM for Android Jelly Bean 4.1.2 (cm-10-20130117-1903-otter-sgt7.zip) on my rooted 1st generation Amazon Kindle Fire. I am very pleased to say that it boots much faster than the stock Amazon-loaded OS (forked Android Gingerbread 2.3)!
Loading another ROM removed the Amazon Marketplace, but does not load Google Play. After loading CM10, Google Apps (gapps) must still be loaded in order to get Google Play. Once this is done, Android apps – including the Kindle reader – can be reloaded.
The problem is that after loading CM10, some apps would not install correctly using Google Play. The Kindle Reader app in particular gave an error that was particularly vague: “Unknown Error code during application install: ‘-24′”.
Fortunately, the fix for this is rather simple. After loading Google Apps, do a factory reset using TWRP or whatever boot loader you’ve installed on the Fire. From what I’ve read, this may apply to other Android devices as well.