While everyone has their own method of cleaning a bore, most do too much brushing and pumping of patches.
Use a good brush with a copper solvent like Hoppe's #9.
NEVER dip brushes or patches in the bottle, this will contaminate it and ruin the solvent......use a plastic hobby paint transfer bulb to apply solvent.
Keeping the brush wet with solvent brush the bore straight through and out the muzzle before pulling it back. Make about 20 passes.
Some people don't reverse the brush, they unscrew it and only run it one way.
This does keep solvent out of the action.
After brushing, rinse the brush in a solvent like paint thinner or hot soapy water to remove the bore solvent from the brush, otherwise the solvent will "eat" the brush up, and they won't last as long.
Replace brushes often since they don't last long even when taken care of.
After brushing the bore, run two soaking patches through the bore.
Don't "pump" the patches up and down the bore since this does little or NO good.
Allow the bore to soak at least 30 minutes.
The "secret" of bore cleaning is to give the solvent time to work.
After 30 minutes or more, run a soaking patch straight through and check it for the blue or green stains of copper fouling.
If you see any, let the bore soak longer and test again later with a soaking patch.
Continue until the patches come out with no stains, then dry and lube.
Another good method is to brush the bore, then plug it with a chamber plug, fill the bore with Hoppe's and let soak over night.
If you have a semi-auto, remember there's a gas port and the solvent will leak out if you fill above it.
Remember, if you "pump" a patch up and down the bore, the patch will get steel stains on it and it looks like the bore is still dirty.