Vikings are White
Norðmenn (Northmen), and thus Vikings, are traditionally and historically white, and the primary factor is biological.
As Norsemen, our skin is white because of environmental adaptation, and namely, evolving in areas that do not see the sun that much, including for months at a time in winter.
Having our skin tanned is not healthy. It is a reaction from our body that signals we have received too much sun exposure and we are endangering our body. Exposure to UVB actually destroys our melanocytes, or skin cells, resulting in first degree burns and what we commonly refer to as sun burns. Exposure to UVA also leads to further damage to the skin visible through accelerated aging. Furthermore, exposure to both UVA and UVB damages our DNA.
The fairer our skin, the less melanin we produce, an essential natural pigment that captures and neutralizes UV's. Whiter skins therefore have less resistance to sun exposure, as well as a higher risk of skin cancer from sun exposure.
The whiter our skin, the less sun exposure we actually require. We in fact only need 10 to 15 minutes of sun exposure per day on average in order to produce enough Vitamin D, which is essential to absorb calcium, promote bone growth, and maintain a healthy muscle mass.
Natural protection against sun damage includes body hair, and thus beards (see "The Benefits of Beards"), or simply living in an environment we, as Norsemen, have adapted to over hundreds of thousands of years of evolution: The North.
In contrast, the skin of people who have not evolved in northern environments is darker, and even completely black in some regions such as Africa. This is an adaptation to near-constant exposure to the sun.
Because dark-skinned people have evolved to thrive in sunny environments, they have no visual mechanism, such as tanning, to signal unhealthy exposure to the sun, as they are more resistant to UVA and UVB exposure.
The darker their skin, the more melanin they produce. Darker skins therefore have much greater resistance to sun exposure, as well as a much lower risk of skin cancer from sun exposure.
The darker the skin, the more sun exposure a person also requires. Inadequate sun exposure, such as living in northern environments, results in Vitamin D deficiency and leads to various health ailments, including but not limited to weaker muscles and bones, increased risk of death from cardiovascular diseases, cognitive impairment, severe asthma in children, cancer, and is also linked to type 1 and 2 diabetes, hypertension, glucose intolerance and multiple sclerosis. In the latest study from the Department of International Health, Immunology and Microbiology at the University of Copenhagen, Vitamin D was also found to be instrumental in activating human T cells, essential to keep the human body healthy overall.
People with darker skins, including those from Africa and the Middle East, have not evolved to live, let alone thrive, in northern environments.
Another reason Norðmenn are white, is cultural. Cultural specificities are indeed normally and generally associated with race. As a result, Viking predominantly bred with the same ethnicities, and even limited raids to settlements of the same race.
Occasional Multi-racial Vikings
According to Landnámabók (Book of Settlements), one of Iceland's first settlers, over 1,200 years ago, is named Geirmundur Heljarskinn Hjörsson. The nickname, Heljarskinn, translates to "skin like hell”. In context, it does not refer to black skin, but instead to a man who isn’t as white as his peers. This is because Geirmundur Heljarskinn isn't completely white. He is half Mongolian, and the product of the union of his father, Norwegian king Hjör Hálfsson, with a Mongolian woman, during an expedition in Bjarmaland (Siberia). Iceland's part Mongolian origins are further confirmed in the country’s gene pool, and explain why some ethnically Icelandic people, such as Björk, have Asian physical features.
Such multi-racial Vikings did in fact exist throughout history, but such instances were exceptionally rare and an exception rather than a rule.