NO WARRANTY
(USE AT YOUR OWN RISK)
Download CF-Auto-Root
https://autoroot.chainfire.eu/
Samsung - SM-N900T (choose your model from the list)
The files will also have Odin 3 with it (it's recommended to use the
Odin attached with the files where it disables unnecessary options for less problems and confusion)
Extract All Files.
Download SuperSU
From the play app store
You should have Galaxy Note 3 USB drivers installed in your PC if not download it here
http://www.samsung.com/us/smart-switch/
Connect your phone to the PC ....
run(smart-switch if USB drivers are not installed)
and turn it off.
After, Press and hold
Power+Volume down+Home to go to download mode.
When In download mode use PA in Odin to use the CF-Auto-Root exploit and flash it.
After that from the SuperSU you could install it to the system from the SuperSU settings and choosing "Install SuperSU into / system" your phone will restart.
Congratulation your note 3 Android 5 is rooted.