You may very well not need to remove the extractor and extractor rod from the cylinder. The yoke or crane should slip off the rod easily, once you remove the yoke or crane—I can't remember which Smith & Wesson calls it—from the frame.
I simply soak the entire assembly in a jar of Hoppe's No. 9 for several hours, then scrub the cylinder face with a bronze-bristled brush, then scrub the extractor with a Nylon-bristled brush, then clean the bores with first bronze-, then Nylon-bristled bore brushes.
If you've got a .38 caliber revolver, don't be afraid to use .40 or .45 caliber brushes in the cylinder bores: they get the job done faster, and won't cause any harm, although they'll wear out a little faster than .38 caliber brushes.
By the time you've got the rest of the cylinder squeaky clean, chances are the extraction mechanism(s) will be clean, too. Times I've taken everything all the way apart, I've found considerably less grit, grime, and crud than expected.
Best of success, eh?