For the body, most people use a unitard. You'll want a mock turtleneck one for most characters, though a few (Coricopat, Tantomile and Gus, for example) have a scoopneck. Or, you can make the costume using a turtleneck and "legging" type pants, or use a leotard and tights, which is a little cheaper than a unitard, and looks more authentic than pants and a shirt.
You can find mock turtleneck unitards at ADA Discount Dancewear, in both cotton/lycra (not shiny, takes dye well, not 'slick' in texture) and nylon/lycra (shiny, will take most dyes well, may not take all dyes).
To color the unitard, you will want fabric markers and possibly floral spray, which many costumers have found will color the unitard well without making it too stiff. If you have an airbrush, you might be able to use other kinds of dye or paint - it's the thin coating a spray gives you that really makes the difference. Remember when you are coloring, put something inside your unitard, to keep the color from bleeding through to the other side. Newspaper or paper towels should work fine.
You can find floral spray at Jo-ann's Fabric & Crafts, as well as many craft/hobby stores you may have locally, including Michaels and A.C. Moore.
You may also want to try Rit dye for large areas - try using a sponge to put it on. You can find Rit dye at most crafting stores, as well as CVS stores (at least in my area). Online, find it at: Rit Dye.
You can find many other kinds of dye and paint that you may want to experiment with at Dharma Trading Co. Be sure to read what fibers each dye will color, and how to set the dye before you buy something. Not all dyes will work on all fabrics, and some setting processes, would be good for the fabric of a unitard, or may not allow you as much control over the color application as you'd like.
Find more information about making Cats unitards at these sites: