I agree with Ken's advice, although I've never found RIT to be reliable. Browns often come out red and the black dye often produces a dyed article that is purple or bluish. If you're willing to spend a little more money, I would recommend trying a dye from Pro Chemical and Dye (http://www.prochemicalanddye.com/home.php). Their disperse, washfast acid, and lanaset dyes will produce a deep, dark, predictable color that won't wash out or transfer to your clothes and it will resist UV fading better than RIT.
I used the PCD washfast acid dye to make a black and white patterned piece of nylon solid black. I was sure the white areas would still be a little visible. They aren't. It came out solid black and all of the excess dye rinsed out in the first rinse (the water was clear on the second rinse). Rinsing out RIT dye is a long process, and you'll never get it all out. It will continue to leach out for days or weeks afterward.