[Fire HD 6 and 7 General] bibikalka:
Update2: Updated for FireOS 4.5.5.1. Note that you can root 4.5.5.1 directly via Kingroot. With root, you could get TWRP to work, and re-install any FireOS 4 or 5 with all the goodies. The instructions to get TWRP working are here (you need to flash 4.5.3 bootloaders if you do not start with FireOS 4.5.3 !!!) : http://xdaforums.com/showpost.php?p=62011272&postcount=2

Update: I had a couple of knee-jerks, but realized that I can make TWRP say "full_ariel" (I've changed the string), which enables the standard Amazon update files to work !!! I did test the modified TWRP (and the attached zips), everything works fine.

This 4.5.5 update provides no perceptible difference with 4.5.4, and everything has been running smoothly for me for the past few weeks. So out of the general principle I do recommend upgrading!

If you are just joining us, and have a pristine Fire HD, head over here to get your fire rooted and TWRP installed [all steps up to and including 3) ] :

http://xdaforums.com/fire-hd/general/root-v2-0-fireos-4-5-4-gapps-t3150613

With TWRP installed, there is a quicker way to install 4.5.5, without changing the update file (in contrast to my earlier instructions for 4.5.4). This is a dirty flash, and will keep all the prior settings and programs (if you have not had GAPPS yet, see additional instructions below):

0) Turn off wireless (airplane mode)

If you used JMZ tool for GAPPS install or did a manual update prevention, clean up GAPPS update blockers by deleting the following dirs in /data/app (this will make your life easier later on!):

com.android.vending-1.apk

com.android.vending-2.apk

com.google.android.gms-1.apk

com.google.android.gms-2.apk

1) Boot into TWRP, flash the attached 4.5.3_twrp_recovery_full_ariel_string.zip. Reboot to recovery (TWRP)

2) While in TWRP (this modified version self-identifies as full_ariel) :

a) flash the respective Amazon ROM 4.5.x update (the latest one?): http://xdaforums.com/fire-hd/general/fire-hd-rom-links-t3173102 (rename to zip before flashing)

b) Must do this (!!!) - flash attached 4.5.3_twrp_recovery_uboot.zip

c) flash attached make_space_v01_fireos4.zip to remove moffice and weather in order to increase space in /system.

d) flash Pico (Uni) GAPPS for KK http://www.androidrootz.com/2015/03/download-android-44x-kitkat-pa-gapps.html [if you did not have GAPPS before, skip this for now, and come back after Step 3) ]

e) wipe cache and dalvik

f) before reboot say yes to root

3) Boot into fireOS (wait for it to upgrade)

a) install SuperSu

b) disable OTA updates [in ES Explorer rename /system/priv-app/DeviceSoftwareOTA.apk to *apk_ ] - The script in make_space*zip already renamed this file automatically, so nothing to do here.

c) install Xposed. Note if you are using 4.5.5.1, you'll need to get the Xposed installer from here in order to address SDK version 19 error :

http://xdaforums.com/showpost.php?p=64063168&postcount=62

d) install Busybox

Optional 4) : if you did not install GAPPS in 2), boot to TWRP, and install GAPPS

That's it! Whatever you had working before, should be working now the exact same way.

If you prefer a manual way where you repackage the 4.5.5 update file (and perhaps remove some apps from /system), see these instructions (the updater-script is the same) :

http://xdaforums.com/fire-hd/general/how-to-upgrade-to-4-5-4-twrp-root-t3141648