Before we get started, there's a few exceptions with the firmware it supports. Most notably, on the KT2 and PW2 (6th generation hardware), there was a firmware update 5.12.2.2 which patches the exploit KindleBreak uses, in which case you should be using WatchThis instead.
TL;DR of this jailbreak: Download the zip, copy the files to the top level storage of your Kindle, go to file:///mnt/us/kindlebreak.html, wait for it to restart then go to Post Jailbreak.
Prerequisites:
Here is a table on which models are supported, and what firmware versions are supported. Unlike WatchThis, any models with any versions on this table will be supported.
Connect your Kindle to your computer, then copy the 4 files above to the root of your Kindle's storage (where the documents folder is.)
It should look something like this.
Eject and unplug your Kindle.
Go to the Experimental Browser from the top right menu, then navigate to
file:///mnt/us/kindlebreak.html
(Make sure that it is file:/// and not file://)
Your browser should freeze, crash and after some time (this can range from few seconds to several minutes depending on your device) your Kindle will reboot. It'll probably give you some kind of error window with the title along the lines of "Application Error" or "Collecting Debug Info".
Once your device has rebooted, you're finished with the tutorial. Head onto Post Jailbreak section to finish setting up things.