You shouldn't use a solvent as it might not be good for the seals, cause them to swell and degrade, etc. There's no powder or carbon to dissolve anyway.
A couple strokes, and I mean 2, with a bronze or even a nylon brush of the appropriate size coated in silicone oil will get pretty much anything out. Then, push a patch that's been wetted with silicone oil through, and your barrel is good. Put one drop -- ONE DROP -- of silicone oil on the gas port at the breech and on the breech face to lubricate the seals. It is necessary to lube th eseals, but excess oil may "diesel" ie ignite when pressurized, and your air rifle will sound like a 22 for the first shot. It would very rarely do any damage but you should try to keep it to a minimum.
Also, try lubing your pellets. When I get a new can, I put a quick spray of silicone directly in th etin and shake it around a little. Shoots more consistently, and leading will be virtually none.