In this article, you'll learn how to root a Kindle Fire. This tutorial is just for the 1st Kindle Fire. It doesn't work on Kindle Fire 2nd gen or Kindle Fire HD.

1. How to root a Kindle Fire
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2. Rooting makes Android OS more interesting. It allows you to
modify the system. For Kindle Fire, rooting is more meaningful as
it runs an Amazon custom Android, not the real Android. If you
want to install some custom Android ROMs on Kindle Fire, rooting
is the first thing you need to do.
This article provides a simple rooting tutorial for the 1ST Kindle
Fire. If you want to know more about rooting other Kindle
devices, Please visit XDA forum.

3. Step1 Get rooting tools and call CMD
Click here to download the rooting tools. Extract the files in a
folder. Hold “Shift” key and right click. Select the option “Open
command window here” to call CMD.

4. Step2 Run ADB in CMD
Type these commands below one by one in CMD first.
adb push fbmode /data/local/fbmode
adb shell chmod 755 /data/local/fbmode
adb shell /data/local/fbmode
adb reboot
Your Kindle Fire will reboot and keep at the loading interface.
Then type these commands:
fastboot -i 0x1949 flash bootloader u-boot.bin
fastboot -i 0x1949 flash recovery twrp.img
fastboot -i 0x1949 oem idme bootmode 4000
fastboot -i 0x1949 reboot

5. The Kindle Fire reboots and the loading interface has changed.
Long press the power button and choose “Recovery” option(press
power button to select different options and tap the screen to
enter). Your device will enter the TWRP.

6. Now continue to type commands in CMD:
adb shell mount system
adb push su /system/xbin/su
adb shell chown 0.0 /system/xbin/su
adb shell chmod 06755 /system/xbin/su
Choose “Reboot” at TWRP on your Kindle Fire and the system will
reboot.

7. Step3 Install Superuser
Copy the Superuser APK file at the rooting tools folder to your
Kindle Fire. Install it on your device. If you can see this picture
when running an app which need administration permission, that
means you’ve rooted Kindle Fire successfully.
Source: http://www.epubor.com/how-to-root-kindle-fire.html