This is a complicated question because not all dark-skinned people do have dark eyes.

The answer, though, is evolution by natural selection. More melanin (the pigment that colors skin) in a person's skin resists more sunlight before burning, and more eye pigment (and thus darkness) does the same for eye protection. Thus people who live in areas with a lot of sun have an evolutionary advantage if they have dark skin and if they have dark eyes, so over time humans have evolved that way.

Skin and eyes are controlled by different genes, though, so if a dark-skinned dark-eyed person marries a light-skinned light-eyed person, the children will have a mix of both kinds of genetic material, and if those children marry similar children of mixed ancestry, the resulting combinations of skin and eye color will be all over the place. This is why there exist light-eyed people with dark skin and dark-eyed people with light skin.

Needless to say, in the modern age when we have sunscreen and sunglasses, these evolutionary adaptations no longer matter so much.