Rit dyes will work on cellulose fibers including cotton but they do bleed in the wash and need to be heated to take properly. A better option is a fiber reactive dye. Procion mx is a good fiber reactive dye and you may be able to find it at an indepent arts and crafts store if you have any around. If you don't, look for Dylon permanent fabric dye (they're in colored pouches): Joann's and Walmart carry dylon and I think a few other big box arts and craft stores do as well. Each packet dyes a half pound to a medium shade.
Fiber reactive dyes do need to be extensively washed out after dyeing: one cold wash and at least one hot wash, but once all the extra dye comes out it won't bleed anymore while items dyed with RIT dye will bleed a little bit each wash. Fiber reactive dye can also be used with warm water straight from the tap and doesn't need to be heated.
Paula burch's site is very helpful and if you have a question that isn't answered, there's even a dyeing forum.