A black eye is a simple case of having a bruise. A bruise is basically a collection of blood that has pooled under the skin as a result of some kind of blunt object coming into forceful traumatic contact with it. This is not something to worry about though you would be sporting a ‘shiner’ for some days. There might also be some pain in the area, which you would need to attend to but overall you needn’t worry about any other health hazards.
When the body suffers some kind of damage, the usually progression of events I blood clotting, clearing of the wound by phagocytes, rebuilding the vascular system and finally replacement of the skin tissue. In your case, apart from the blood clotting and rebuilding of the skin tissue, every other process would continue as normal. You would have to use a combination of warm and cold compresses to heal this wound and get rid of the black eye. The cold compress should be used to reduce the swelling and inflammation. When this reduces, start using a warm compress to improve the circulation and heal the area. Since most of the damage done is under the skin, there is little in the form of intervention that you have to undertake.
If your reference to a black eye is, however, related to hyperpigmentation of the skin around the eye, then the treatment is completely different. Hyperpigmentation is a result of too much melanin, the skin pigment, in the skin around the eyes. This can sometimes happen as a result of a condition called melasma, though it rarely happens around the eyes. What you need to do is to start exfoliating and bleaching the area. Exfoliate your skin with a salicylic acid face wash. This works by breaking down the binding skin proteins making it easier to simply wash off. After this procedure, take a vitamin C rich mixture and apply it to the skin to bleach it. Some effective sources of vitamin C in nature are lemons, onions, and strawberries. Since these can be quite cumbersome to create at home, you should be able to get a strawberry extract face cream or face wash, which will also serve your needs well. From a dietary perspective, it is important to increase your intake of vitamin C from citrus fruits, azelaic acid from wheat and rye, and vitamin E from seafood and leafy green vegetables.