Here are excerpts from a document I put together for friends and family who are getting started in firearms.
Materials:
Go to a gun store or a department store with a sporting goods/hunting/firearms department and buy some foaming bore cleaner. Outers & Breakfree make a couple of the better formulations, but all of them seem to work pretty well. Also buy some Outers Nitro Solvent (or other generic powder solvent). If you don’t like, or can’t tolerate the odor of typical solvents, Hoppe’s Elite makes a gun cleaner with a very mild odor as does M-Pro 7.
You’ll also need a brass or bronze bore cleaning brush and a cleaning jag (patch pusher), both suitable to the caliber of your handgun. Get a sturdy cleaning rod, a one-piece rod is always best. It’s handy to have two cleaning rods; put the bore brush on one and the patch pusher/jag on the other. It saves time since I find that it’s efficicient to alternate the use of those two items during the cleaning process.
You can buy or make cloth bore cleaning patches, but a strong paper towel (like Bounty or the blue “shop” paper towels) will work for cleaning patches. Trying to use a typical, flimsy paper towel is probably a lost cause—it will tear too easily. Just tear off a piece of paper towel and fold it over once or twice so that’ it’s roughly 2”x2” or smaller. If you make your patch too big or too thick it will be very hard to push it through the bore and it could get even get stuck. Start with a single fold-over and a small patch to get a feel for how much force you’ll need to get it through the bore. You can always make the next patch a little bigger or fold it over again, but if you get one stuck it’s a pain. When using paper towels you may have to experiment a bit to get the patches started so they don’t immediately tear.
You’ll need some safety glasses. Any time you're using solvents and bore brushes you'll need to protect your eyes from flying droplets of solvent and bristles that sometimes liberate themselves from the brush.
Spray some nitro/powder solvent into the bore and brush it out to remove the loose fouling. You’re not trying to get it the barrel completely clean on the first try, so a few passes of the brush are enough. Push the brush all the way through the barrel before pulling it back through. If you try to reverse the brush in the barrel it tears up the brush—in extreme cases it can cause the brush to bind which makes it very difficult to remove.
Now push a couple of dry patches down the barrel to remove the dirty solvent & loose fouling. Next spray some foaming bore cleaner in the bore and set the barrel aside. If the gun is a revolver you can set the whole gun aside for awhile until the bore cleaner does its work (read the directions on the bore cleaner you’re using). If you have a semi-automatic then you can clean the other parts while you leave the bore cleaner working.
After waiting an appropriate interval (per the bore cleaner instructions), run a clean, dry patch through the barrel to push out the solvent/cleaner & dissolved fouling. Put some nitro solvent on a bore brush (or spray it into the bore), run the brush through the barrel a few times and then follow with a dry patch. Spray the bore full of foaming bore cleaner and leave it until the next commercial break on TV (10 or 15 minutes ) and then repeat the “nitro solvent/brush/patch/bore cleaner/patch” steps until the dry patch after the foaming bore cleaner comes out clean, not brown/black, blue or green. Now the bore is clean.
Be aware that small bits of the cleaning brush will wear off in the bore and if you leave the debris from the brush in the bore before spraying in the foaming bore cleaner you'll always get gunky/blue/green patches since the brush material (bronze or brass) contains copper. That’s why you always patch out the bore after using the nitro/powder solvent with the bore brush and before you apply the foaming bore cleaner.