Easily Root Kindle Fire 6.2.1 Using Kindle Fire Utility
The 6.2.1 update made things difficult for those that wish to have a rooted device by breaking a popular method of rooting the device. The Kindle Fire Utility makes the process simple and allows the SuperOneClick utility to once again function. Once rooted, you can once again use your custom wallpapers (without them changing) and install missing framework files, etc.
Step 1 - Download the Kindle Fire Utility zip file.
Step 2 - Unzip the downloaded file.
Step 3 - Connect your Kindle Fire to your computer using a USB cable.
Step 4 - When the Disconnect button appears on the Kindle Fire, tap the button to get out of USB drive mode.
Step 5 - Go back to your computer and open the Kindle Fire Utility folder. Double-click install_drivers.bat to install drivers to allow the device to properly communicate with your computer.
Step 6 - Once the drivers have properly installed, double-click run.bat to start Kindle Fire Utility.
Step 7 - The Kindle Fire Utility window will give you 4 different options, select option #3 (Install Permanent Root with Superuser). The program will go through the process of rooting your Kindle Fire and rebooting the device. Once the device has come back up, you now have a rooted Kindle Fire.

Kindle Fire Utility