There is no compelling reason to clean the barrel, so don’t.
Because airguns operate with a blast of air, most small particles of residue are blown out of the barrel as the airgun is fired. Occasionally, the bore should be swabbed out with a clean, dry patch. I use no solvents on airgun bores because of the danger of the solvent getting into the valves. The valve seats are usually made of some sort of plastic or rubbery material that can be damaged by cleaning solvents. Just push a dry patch through the bore a couple of times to remove any small particles of lead or other material. Do not allow any gun cleaning solvent to enter the pump mechanism or cylinder compartment of an air or CO2 gun.
Really Long Answer for the self-proclaimed expert, that won’t accept the good advice in the first two answers
Here is some more specific advice:
You generally do not clean an airgun barrel. One initial cleaning when the gun is brand new, may be all you ever need. It is better to under clean, than over clean. Depending on your equipment you can damage the rifling or more important the Crown (the very end of the rifling, as you exit the barrel).
The only reason to clean a barrel is if you feel the accuracy has been reduced from what is normally expected.
Do not use an uncoated steel rod or steel brushes in inside of a steel barrel, a brass rod is OK. If it is a Brass Benjamin or Sheridan, some type of plastic “pull though” fishing line or use a wooden rod to push a patch through. A one piece aluminum cleaning rod may also work. There is even less reason to clean a Benjamin, just leave it alone.
Use only a clean, dry “cleaning patch” of the proper size.
All of our PCP guns that have a magazine have an internal O-ring in the barrel; Care must be taken to not damage this 0-ring. I only use a” pull though” type and start at the muzzle end. Only use a dry patch.
What not to use: Do not use any firearms gun cleaning solvents. Do not use Silicon oil.
Some have reported good results using Ballistol or Simple Green as a cleaner. I have not tested this on all our guns so I am hesitant to recommend it, best to stick with the dry patch.