DNA do effect our outer appearance. However, the idea there are multiple race of human today, is actually a common misconception. A lot of people think people of different skin color are of different race, that is inaccurate. The scientific term for race is "breed". The scientific definition to qualify as different breed, is to have genetic mutation that as a result, turn one breed into two breed, these two breeds have different but very similar DNA. Because their DNA are so similar, they can continue to be produce off springs, which can then produce further offspring and so on.
Amount all homo sapians, there are no genetic mutation, disregards to whether you are white, black, Asian or native American and so on. What cause people of different places to have different skin color is same as why, people have different physical appearance, some people have round face, because their DNA dictated it, other have square face because their DNA dictate it. A person's skin color is also dictate by a series of DNA, ranging from color, shade etc. However, due to human live all over the world, it is found people of different skin tone, better adapted to different area. As a result, eventually, people of all places, not only have different skin tone, but different skin color. African people usually are black, because black reduce the risk of skin cancer in that very hot climate. So, in the end, only those who have black skin remain in Africa, other color skin DNA graduetly disappeared. in Europe, white become camoflauge, so with every passing generation more people with lighter skin tone emerged, some of these people then move to Asia, because Asia is warmer, so the white camoflague is useless, so, Asian people stop getting lighter, while European keep getting lighter in skin tone. If you pay attention, you realized, white people and Asian people actually have the same skin color, but Asians are dark yellow, white people are pale yellow. And Asian hair are in fact not black, but dark brown or dark red, but it is so dark, it look black, but you hold it against a sun, you realized it is either brown or red, not black.