Android 4.1 Jelly Bean is the latest OS of Android, Google Nexus 7 has in-built Jelly Bean.Thanks to XDA developer Hashcode,who has released Android 4.1 Jelly Bean based custom ROM for Amazon Kindle Fire.The ROM is still in developmental stage so you might experience bugs but will get fine with the release of newer versions.
If you want to download and install Android 4.1 Jelly Bean on your Kindle Fire then follow the steps mentioned below-
1. In the first step backup data such as SMS and call logs, contacts, internet setting, apps etc of your device.
2. Download the latest version of ROM to update your device from here and Google Apps from here and download the .zip file into SD card of Kindle Fire.
3. Install a custom recovery,TWRP recovery installed on your device.You can enter into recovery mode by turning off your device and pressing and holding Power button for few seconds.
4. In this step you will have to create a backup of your existing ROM which you can do by clicking on “Backup” to option to make a backup of the current ROM and then go to Main Menu.
5. Select Factory reset to do full wipe of data.
6. Go back to Main Menu and then select “Install” and choose .zip file of Android 4.1.1 Jelly Bean ROM.
7. Select “Install” and choose .zip file of Google Apps Package.
8. Once the installation of Google Apps Package is done click on “Reboot system” to reboot your device.
You will be now able to enjoy Android 4.1.1 Jelly Bean on your Kindle Fire.