Actually this is a part of my current thesis! So it's important to note that Vikings aren't a homogeneous grouping, they lived in different areas and groups with a few shared customs and religion between them, but depending on where they were from they had different norms. So the logical first place to start is written accounts, because bodies degrade. This means we can't usually speak to hair, skin, etc.
Archaeological studies to tell us a few interesting things about their appearance. Many Vikings would carve straight grooves into their teeth and occasionally fill them with pigment, we don't know why they did any of this though. Ibn fada a Muslim traveler notes that some of them did have tattoos but doesn't go into painful detail about location and type, but we do know they were there and likely gendered due to his accounts.
We do know however that the pigments we're often used to adorn not only clothing but the skin, this is noted in some parts of the sagas, especially the sagas of Ragnar Lothbrok. It does mention using it as an eyeshadow of sorts, but for utilitarian uses the sagas mention them using dark pigments much like modern football players do to detract from the glare.
Speaking further to appearance some Vikings were the personal guards of the Emperor of the Byzantine emperor who would make them look more "savage" to dissuade enemies. In their usual appearance they are noted as looking remarkably similar in dress and style to the Anglo-Saxons of the same time period.
But hygenically the Vikings were probably the cleanest people in Europe at the time. As part of their grave goods they are often found with nail trimmers, ear cleaners, and other cleaning implements and are noted for their "unusual propensity" to bathe around once a week at a minimum. These habits are noted in the Icelandic sagas. They were also known to be impossibly vain about their hair as it was part of their pride.
We do know that they were so keen on hair upkeep from the few hair remenants that remain in burials as well as the combs themselves that have been found. As to facial hair many adornments have been found that could be used as either hair or facial hair, but placements within burials suggest some were used for facial hair.
The truly important thing to know however is that a "Viking" is not a people. The people were Norse the act of raiding was called going Viking and so were a very small number of the total people present in the culture and are a poor representative of larger cultural trends. For those you have to look at Scandinavia, Iceland, Ireland, and Scotland (especially the islands where they set up their own kingdoms and wrote the Orkneyinga saga). But overall they are remarkably known to have looked like the Angles and Saxons of England at the time in terms of style and dress with a few oddities such as the tooth carving thing.
I posted a very short notice before in this similar question thread, together with the links to the relevant posts.
The following is a excerpt from my post, mentioned above:
- Some of the male Scandinavians (at least the people in now Schleswig-Holstein) already practiced braids as Tacitus described ca. 2000 years ago, but we don't have enough indigenous sources concerning the contunuing trend of this hair style during the Viking Age itself.
As for the long beard style in Viking Age iconographic sources, Gotland picture stones are probably the only ones:
As for the tattooing among the Norse people, the description of Rus' (either ex-Scandinavian descent of the Russian or Scandinavian) merchants that Ibn Faḍlān met in middle Volga region in the beginning of the 10th century (inspiration source of the notorious Viking film, the 13th Warrior (1999)) is often cited one, as following:
'I saw the Rūs, who had come for trade and have camped by the river Itil. I have never seen bodues perfacts than theirs. They were like palm trees. They are fair and ruddy. They wear neither coats [qurtāq] nor caftans, but a garment which covers one side of body and leave one hand free. Each of them carries an axe, a sword and a knife and never parted from any of the arms we have mentioned. Their swords are broad bladed and grooved like the Frankish ones. From the tips to his neck, each man is tattooed in dark green with designs, and so forth'.
Translation above is quoted from: Paul Lunde & Caroline Stone (trans.), Ibn Faḍlān and the Lands of Darkness: Arab Travellers in the Far North, Harmondsworth: Penguin, 2010, pp. 45f.
This contemporary portrait of King Cnut the Great of Denmark as well as England, in the Book of Life (Liber Vitae) of Hyde Abbey, Winchester (Stowe MS 944) (ca. 1031) is also very famous one.
On the other hand, AFAIK no confirmed contemporary account for eyeshadows.
[Edited]: Italicalizes some proper names.
