I'm sure the drummer guys, like Greg, have it down to a science...but I wanted to just toss out my own process, because even though I'm not a drummer, I have the studio kit and I like to keep the cymbals clean.
OK...so since I bought a lot of used cymbals over the years in order to find what I liked, and then I sold the ones I didn't...most of them came with smudges, oxidation and some with a heavy patina.
To remove all that grime, I just used Brasso and a green scrubby, which does a great job, but leaves it's own black grime (oh, and it WILL remove any printed lettering if you push it too hard...so work around that stuff if the lettering is imporatn to you)_.
After the Brasso, I did like what Greg said above...warm water and soap, and that removes all the Brasso gunk and leaves a nice shiny cymbal.
Now here is where I added my own twist to the process.
My drummer complained how even after cleaning cymbals, as soon as you grab them, you leave new fingerprints.
So I thought about it for a bit, and on a hunch, decided to put down a coat of car wax on them.
It worked out quite well.
The car wax resists new fingerprints, and it has kept the cymbals shiny for a long time now, and free of oxidation....and it's been awhile.
I think I just used your basic Turtle Wax on them...the white cream paste. One coat, buff it off with a dry rag, and done.
There was no negative effect on the sound of the cymbals that I could tell.
YMMV....