My manual procedure has changed a bit over the years, but at present the way that works for me is to soak the balls in the sink in some warm water and a small amount of regular dish-washing liquid. Then I remove each one from the sink and, whilst still wet, run it over with a "Mr. Clean Magic Eraser" cleaning pad.
Then I put the balls on a towel and put a blob of the Aramith Ball Cleaner and spread it all over the surface with my fingers. (Wear rubber gloves if you have sensitive skin.)
By the time I've done the last ball the first ball will be dry and ready to wipe off the polish with a rag or micro-fiber towel. Nowadays I have a BallStar, so I just stick them in that, but as other posters have indicated you can build a ghetto version yourself if you feel the need. (With the Ballstar I only do this full process once every few months, I use the automatic Ballstar cleaning cycle after every session.)
However despite using both the Aramith hand polish and the Ballstart automatic polishing I still have problems with chalk marks on *SOME* cue balls. A constant criticism of mine (and some others) is the relative inconsistency of the surface finish (and often weight) of Aramith/Saluc cue balls. At the bottom of the list has to be the "Measles" Pro Cup which I just cannot get clean no matter what I use. Next to that is the "Aramith Logo" that comes in the Super Pro set (It appears identical to the Pro Cup in every respect except the red spots) and then the Blue Circle that comes with the Brunswick-branded Centennials. By far the best is my 10 year old Red Circle that came retail packaged and sold separately from any ball set. Unfortunately despite this still having the best surface finish/quality compared to any of the newer balls it has now worn down to the point that it is undersized compared to the object balls so has to be retired. I could order a new one from Muellers but I have absolutely no idea if one made today will be of the same quality and worth the effort.