Bore cleaning gets down to personal likes as far as cleaners/solvents as well as equipment. There are also as many different methods as opinions on method.
What I personally recommend in our classes is first get a good one piece cleaning rod, I prefer the Dewey rods myself. Next get a good bronze/phosphorus brush, as well as jag, never use stainless brushes. Cleaning pathces are a preference as to which brand. Solvents are also a preference, but some that I use are good old Hoppe's-9, KROIL, Cleansoil, Butch's Bore Shine, or Shooters Choice. For the Chamber area get a good chamber brush and my preference is to put this on a small one piece rod and chuck it in a drill.
Wet the bore/chamber down with your preference of cleaner/solvent. With the chamber brush on chucked in a cordless drill, insert slowly and rotate slowly only a few, about 3, turns to remove fouling if necessary.
On the bore, brush several times always from the rear, and not the muzzle end, pushing all crud forward using the one piece rod with brush dipped in solvent. Patch from rear forward also until patches come out clean. Follow up with a little denatured alcohol to remove any solvent or oil left in the chamber/bore area, and now you are good to go.
If you are needing to remove copper fouling then wet patch in the copper solvent and let sit for no longer than 10 minutes and then dry patch out. Wet with solvent and then follow up removing any solvent with denatured alcohol to remove all solvent/oil. You do not need to scrub with the copper solvent, as it is the chemical reaction of the ammonia that will loosen the copper to be removed on a patch.
I hope this helps, but keep in mind that it is personal preferences as to what you use and you will develope a routine method that you get comfortable with.