Use a copper or nylon brush with a good rod and a solvent. I like Hoppes #9. Run it down the barrel 5 times and then let it soak for 5 minutes to dissolve copper. Then run it down the barrel 5 more times. There are better copper removers but this has always worked for me. Stay away from anything with over %25 ammonia. It's not good for stainless. Then use a soaked patch until it comes out clean. Finish up with dry patches until dry.
Bore snakes are good for debris but are about useless for copper, which for me still makes them about useless.
Just my .02