There are a few remedies that people claim will help lighten dark skin, but I have not tried any of them so I can't promise any results.
One remedy is to rub lemon slices onto your skin. Lemon is thought to act like a natural bleaching agent. I have also seen suggestions to mix lemon juice with yogurt and apply that to the skin.
Another idea is a teaspoon each of lime juice and cucumber juice mixed with just a small amount of turmeric. Apply to skin and wash off after 20 minutes.
It has been suggested that dark underarm skin is caused by the rubbing of tight clothing and shaving. If that is the case, wearing sleeveless or looser clothes may help as well as waxing the underarms instead of shaving them (depilatories are said to be as bad as shaving).