This is a time-consuming project, but it is worth the effort if you really need a specific color of wig and can't find it. I'm going as Kim Boggs from Edward Scissorhands for Halloween, and she has this specific style and color of wig that I just couldn't find anywhere...even online! So I started researching how to dye synthetic wigs and learned that you can't paint them; it makes them stiff, you can't use normal hair dye or RIT dye because it is intended for natural fibers and won't stick to the wig, so really the only way to (inexpensively) do it is to use permanent ink. Thought it would be nice to put up a short tutorial, I could definately have used one when I was figuring out how best to do it. I didn't decide to do a tutorial until I was halfway done, so I don't have photos of what it looked like before...But just imagine platinum shiny.
(NOTE: If you're doing a short to medium length wig it won't take you 5 and a half hours, this wig was pretty long. This method would also be fine to just add color highlights, lowlights, etc.)
The first thing you've got to have is the wig. I wouldn't waste the time and effort of coloring a wig unless it is of OK quality, get one that you try on before you buy, if you can. Not one of the really really cheap ones that come in a plastic bag all scrunched up. My wig was about $40...more than I originally intended... but you want it to look good, right?
Before you start coloring, test your Sharpie on some hair under the wig (from the part that will be against your neck, in the back). Then, start coloring.
Take a small section of hair at a time and color in light, gentle strokes with the side of the Sharpie. You may want to hold it at a slant. Start at the "scalp" of the wig and work your way down.
Don't worry if it doesn't seem dark enough; you'll want to start out light and go back to darken it later, if you need to.