There are two major methods of cleaning carbon out of the AR: Mechanical and chemical.
Mechanical methods use a scraper tool. The pictured C.A.T. tool shown is one of the best, because unlike most tools, this one tool does it all.
The chemical method is to use a carbon removal chemical like Slip 2000 Carbon Killer.
All that's needed is to shake the jar thoroughly then soak the parts in it for 15 minutes. Finish by brushing the areas.
For some areas, you can wet patches and pack them in the carboned area.
DO NOT, EVER put anything in the gas tube. The tube is totally self-cleaning and putting anything in the tube is very much a NO-NO.
If you ever have any kind of trouble with the tube, you simply replace it.
The only tube cleaning that even might be needed is to wipe off the outside of the tube where it enters the receiver.