First buy BRONZE bore brushes in bulk from Brownell's.
Bore brushes get "used up" fast by friction and the bore solvent. When they start feeling loose, pitch them. To make them last longer, wash the solvent off before putting them away.
To clean, use bore solvent and a good brush to make about 20 passes through the bore, then one or two soaked patches.
Allow to soak for at least 30 minutes, then soak a clean patch and run it straight through the bore and out the end. If the patch has blue or green stains, the bore is still copper fouled. Allow to soak another 30 minutes or so and continue until a patch comes out without stains.
Some pointers:
"Pumping" a patch up and down a bore does nothing. A patch is just to carry solvent in and out of the bore.
If you pump a patch up and down a bore it gets steel stains from friction. Many people think this is fouling and complain that no matter how much they clean the bore is still fouled.
To remove lead, buy a Lewis Lead Remover kit from Brownell's. This is a tool designed to remove leading WITHOUT damaging a bore. It's been around since the 40's.
Watch the video too:
http://www.brownells.com/aspx/NS/store/ProductDetail.aspx?p=21587&title=LEWIS%20LEAD%20REMOVER
After using the Lewis tool, brush the bore with a bore brush to remove the last remaining lead in the corners of the rifling.
Some shooters get SOLVENT PROOF tubing and make a long narrow tube that will hold an auto pistol barrel. They brush out the bore and run one soaked patch to remove the filth, then put the barrel in the tube and fill it with Hoppe's Number 9 bore solvent and let soak overnight.
Pitch the used solvent.
Usually, this will remove any remaining copper fouling.
You can use patches made from a lead removal cloth, and they do a good job, but they still don't get the leading out of the corners of the rifling. You still need to brush.
NEVER use stainless steel bore or chamber brushes. These are for gunsmiths to use on bores and chambers so bad, you have nothing to lose.
Use a hobby paint and solvent transfer bulb or a large eyedropper to apply solvent to patches and brushes. This prevents contaminating the bottle of solvent.