Here is my traditional viking build. If you're looking for the badass unarmed viking build, you want Robbaz, King of Sweden. Skyrim is pretty much a love letter to Nordic culture, so what better way to play than as a Nordic viking? Viking, incidentally, was a job description rather than a nation or people.
Traditionally, the spear was the most common weapon wielded by vikings. That's not an option in Skyrim so your best bet would be axes. Every household had an ax for chopping wood and other chores around the farm. They were small, light, but devastating. The ax had cutting power but was wielded more like a blunt force weapon such as a mace. They could hook onto the back of a shield and pull it away to prepare for dealing your enemy a death blow. Swords were a sign of great wealth and power in Norse society. Because it cost so much money to forge one, swords were made to last and would often be passed down through the family for several generations. If you're role playing a viking, you could use an ax for the lower levels and switch over to the sword after you've become thane of multiple holds and own several houses.
Vikings favored mobility in combat so they wore light chainmail and carried a shield. I like wearing fur or leather armor until I can get up to scaled and finally dragon scale armor. Elven and glass armor look fine if you're going for a noble warrior type, but you're playing a savage viking! Wearing fur armor also gives you an incentive for keeping your Block skills up to par because it's going to hurt a lot if the big beefy two handers land a blow upon your person. In real life, two handed battleaxes and warhammers were created as a response to the development of full plate armor. It was exceedingly difficult to make a killing strike with a sword against plate armor, unless you lucked out by getting in on one of the joints. Blunt force trauma was more effective against armored opponents.
As always, there are enough perks left over before the soft level cap of fifty for you to invest as you see fit. I dropped some into the Speech tree, because why not? You'll eventually have more gold than you know what to do with no matter how you play, but this way I was insanely rich before I even killed my first dragon.