If the genes that control hair, eye and skin color are preserved in the DNA sample they use those colors.
Otherwise they make their best guess based on other data, imagination, or just plain prejudice.
It is not just the hair and skin color that I wonder about.
The hair color is, I think, the most straightforward to determine from the genes. Skin is not as straightforward, since many non-genetic factors affect color. But what I really wonder about is the assumption that Neanderthals were mostly bald like modern humans. I’m not certain the genetics of the pattern of our fur covering is very well understood. And the most closely related primates are almost completely covered in fur. Yes, I know that humans are not really bald, but most of our fur is pathetically thin and sparse.