My personal Instructions
Im sure everyone has a different way of doing things,and perferences.But as of 7-28-13 this just works without any flaws
Kindle Fire Utility v0.9.9 Installation Instructions
1) run "run.bat"
2) choose Option 2) install root with superuser
3) let the kindle boot to the following states
-kindle fire (orange)
-kindle fire (blue)
-TeamWIN logo (TWRP)
-Kindle fire (orange)
4) unlock the kindle inorder for it to go into USB mode to install superuser.apk
SUCESS Now rooted
Now Since TWRP Team doesnt have access to a kindle the recovery sucks, its full of bugs
As of 7-28-13 TWRP 2.6.0 cannot restore the kindle to stock state with the update.zip, and it cannot reliably install a custom rom.These are known issues that were causing problems before this version.I will not debate them.If I follow the instructions to a letter and they dont work more than once its not ready for prime time.SO I suggest installing COTR.Its not as pretty but it does the following flawlessly backups, stock restore of update.zip, and installation of custom roms
Run the kindle fire utility again and choose option #4 to install COTR over the TWRP recovery. Reboot the kindle when the kindle logo with (blue fire) pops up hold the power button till a menu pops up ppress it again to choose --recovery--. Thisstarts COTR, calibrate the touch screen this sets up a COTR folder with settings. Choose "nandroid" --> "backup" to backup your kindle rom and all its apps (optional). Then choose "boot android" since my usb driver wont work. once the stock OS booted I opened the kindle drive and copied my backup from the COTR Recovery Folder ,and I deleted the TWRP folder. I deleted it to save space. Now your ready for installing a custom rom.This is the one I used
http://xdaforums.com/showthread.php?t=2103278
Installation
-reboot into COTR recovery
-Wipe data, cache, system, and dalvik cache
-choose zip flashing and choose rom.zip
-Install the latest gapps for CM10.1.x
http://goo.im/gapps
-Reboot