This is a copy/paste of my reply on a prior thread on this topic. Most of it is relevant to your question but I’ll start by confirming what others have said about using a polishing compound as opposed to paper of some kind.
I prefer to use a cleaning rod with ball bearings in the handle to allow the rod to rotate and follow the rifling, working in a linear fashion to help get into the inside corners of the rifling where lead is more likely to build up. Some folks like to spin it and they get good results too but I’m not a fan of that approach.
I do it to every new barrel I touch. The idea is to smooth out the microscopic surface fretting left behind from manufacturing that tends to abrade lead. More strokes at the breech end and progressive fewer as you get nearer the muzzle. Something like 100 strokesis a good start…25 strokes to the first quarter of the barrel’s length, 25 more to half of the barrel’s length, 25 more to 3/4 of the barrel’s length, and then wrap up with 25 full length strokes. Do not overshoot and let it exit the muzzle because you don’t want to develop a bell mouth at the crown. Then clean the barrel and reinstall it and test.
In my experience, once you get a bore nice and smooth, you will not have to clean it nearly as often and it will be easier to clean when you do.
Like most folks, I use J-B bore compound and then do a finishing pass with J-B Bore Bright. Other compounds will work fine though, you just don’t want something extremely aggressive.
I cover this topic and a few other DIY accurizing suggestions in this article:
https://www.gatewaytoairguns.org/GTA/index.php?topic=130555