By looking at your avatar, you appear to have a darker complexion. Be very careful with skin bleaches as they may end up leaving your underarms a lot lighter than the rest of you!! Before you do anything else, try to exfoliate the area and decide a new way of hair removal besides shaving.
I used to have dark underarms as well, but it wasn't actually my skin that was dark. The tiny hairs that were growing in their follicles showed through my skin and made my armpits look dark gray. No matter how close I shaved them, there was no way to get to the hairs under my skin. Instead, I switched to using an epilator, which is a contraption that rips the hair out by the roots. The first time I used it, it hurt a lot.... I won't lie to you. But every subsequent time I used it, the pain was less and now I barely feel it at all. Since the new hairs arent as deep as the older ones, they come out much more easily. I personally prefer this method to waxing, but I imagine that waxing would give you the same result. However, it is expensive to get it done in a salon and its really hard to wax your own underarms at home (unless you have someone who is brave enough to help you). I bought my Remington epilator about 4 years ago for $30 and it still works great!