I keep a quart jar 1/3 full of Hoppe's #9, when I get home from shooting I pull off the cylinder and crane and drop them into the jar, and wipe some Hoppe's on the cylinder window areas and run a wet patch down the bore, letting it stay wedged into the forcing cone. I let everything sit for 24 hours or so, then clean normally. Most of the stuff wipes off, and if there's any lead in the barrel it pushes right out with a tight patch. I have a couple Lead-away cloths but rarely use them, the front of the cylinder is the only place they're needed, and a brass-bristle brush will take the buildup off with a little elbow grease. You won't need a Lead-away cloth unless you're actually shooting a lot of lead.

Make sure you get all the grunge out of the cylinder's chambers........38 Special ammo (especially the dirty stuff) leaves a ring of crud that will keep a 357 round from chambering. Use a plain brass 357 case with the neck flared, just small enough to fit into the chamber. It makes a good scraper, and won't hurt the gun, as long as you don't use nickel cases.