Because root login is disabled on the serial console as sold by Amazon, some steps are required in order to re-enable root access.
For how-to regarding opening the kindle case, refer to the teardown videos on the web. Hardware access to the serial port connector is required for this method. Familiarity with the GNU/Linux boot process (uboot) is also required.
Because hardware hacking is required, this method is not considered a 'Jailbreak'. It does yield complete access to the kindle kernel and userland.
See full logs and instructions here http://88proof.com/hardware/kindle-20131009.html