Some good advice from others. Here is how I clean my SS revolvers.
Run patches soaked with Hoppes #9 thru barrel and chamber holes, let soak, run another and scrub in and out, careful not to drag the rod across the muzzle end. Follow that with a brush dipped into the Hoppes, give that a few in/out cycles, then another soaked patch. Now hit it all with dry patches till all traces removed of solvent. Now take a couple clean patches and use a Copper solvent, such as Shooters Choice and soak the bore and chambers. Let this sit for around 15 - 20 minutes and wipe dry with clean patches. DO NOT let this sit in the barrel/chambers for long, it can etch the steel (maybe not so much on SS but blued or nickled guns for sure) be sure to wipe dry ANY over spill or run off from the gun, just get it in the bore, copper solvents will ruin blueing or nickle.
Follow up the copper solvent bath with another of Hoppes #9 and dry again.
After the above take the aforementioned Lewis Lead cloth and cut patches from it, vigorously scrub the bore and chambers with a patch or three of the lead removal cloth. Now repeat the Hoppes bath, leaving out the brushing.
Check the bore and chambers and repeat untill happy.
The Hoppes will remove powder fouling and some leading from the barrel/chambers and the copper solvent will do the same for any build up of jacket material.
As for the fouling on the face of the cylinder, the Lead removal cloth works great there as well, much elbow grease and patience.
I've had good luck on the face of the cylinder using Flitz or another mild metal polish.
DON'T let the copper solvent remain in contact with the finish on blued or nickle guns, not even a little, I can't stress enough just how bad this can mar the finish on these guns.
DON'T let the copper solvent sit in the barrel for longer than 15 - 20 minutes, it is ammonia based and can affect the bore/chamber.
DON'T use a lead removal cloth on blued or nickled guns, it too will ruin a finish.
DO follow up this cleaning method with a light coat of oil in the bore and chambers, protect the bare metal, of course wipe this out before shooting again.
I'm kind of odd in that I actually enjoy cleaning my guns, gives me time to meditate and think about things.
Hope this helps.
RD