tl;dr - A monstrous warmongering Civ, but with a few twists to it, revolving around the Berserk UU.
The Vikings are Militaristic and Seafaring, starting with Pottery and Warrior Code, and their UU is the Berserk, which replaces Lonbowmen with a 6.2.1 ampibious unit. And there you have it all - one of the most deadly warmongering Civs in the game. On Pangea maps the Vikings are fearsome, on Continent maps they are a terror, and on Archipelago maps they are bloody hell-spawn.
But let's go point by point, first the traits, then the UU, around which the Viking play revolves much more than that of most other Civs. There's not much to say about Militaristic. In my opinion, it is the weakest of all the traits, since it only shines when warmongering (and even then, rather dimly), limiting your options. But if Militaristic feels just right for one Civ, it's certainly the Vikings. Seafaring is better, it even provides the chance to play the Vikings as something besides pure vicious warlords - if you manage to resist the latter, somehow (I will soon show you that this is not easy, even for a generally science oriented, trade reputation concious, let's-try-to-be-friends-with-all, anti-warmongering wuss like me). The higher commerce in coastal cities and the additional movement point for ships, coupled with the coastal start, can result in some really decent early trade and gold amassment (more so on Archipelago maps than on Pangea maps, of course, but even on the latter the trait is not entirely useless). That could be used for a peaceful playstyle, but with the Vikings, it actually augments the warmongering palystyle much better. So let's finally stop the pretence and take a look at...
...the Berserk! This UU, in and of itself, is madness. Together with the Viking traits, it is pure cataclysm. The fact that it replaces the Longbowman, of all possible units, still reduces me to incoherent giggle fits now and then. Let me tell you why. The Berserk is 6.2.1, so it is right up there with the strongest attack units of the entire Medieval Age, the Cavalry and its UU replacements. However, it comes significantly sooner, with Invention! If you beeline for it (which I strongly suggest, although the term "beeline" might be an overstatement, considering that it's only three techs), some of your opponents might actually not have finished ugrading from Spearmen when your Viking hordes descend upon them. But even if they have, the 1.3.1 Pikemen do not pose a much more difficult problem. Even the 2.4.1 Musketmen, which you will have to confront a little later, are no match for the 6-attack Berserk, when deployed in force. True, its movement is only 1, and one could easily make a point out of stating that Knights and, especially, the UUs that replace them, are still stronger (as a general rule, I would rate movement over attack any time myself), but 6 attack points are 6 attack points. At this early a date in the game, that is double that of the best defender - for dozens and dozens of turns, usually. Find me a UU that generates a similar disparity (I spare you the effort: the only one is the Ottoman Sipahi, and their window before the next stronger defender is much shorter, although their movement of course makes well up for that). So even when deployed on land, in the cover of a few Pikemen, the Berserk will tear through your enemies. The amphibious option of course renders all discussion about movement moot, especially on Continent and Archipelago maps.
The downside of the Berserk is its cost, 70 shields is quite steep (although, to be honest, anything less would be completely broken). But here the Viking traits and the fact that the Berserk replaces Longbowmen come in. Ridiculously, since it replaces Longbowmen, you can upgrade Berserk from Archers (a unit you will produce in the Ancient Age as your mainstay). The cost of this is 150 gold per unit. That is also a steep cost, granted, but you will be one of the first at Invention (the AI does not often beeline for that with a high priority) and you can thus usually snatch Leonardo's Workshop even on Emperor and above. So now the upgrade is 75 gold. Still too much, you say? Well, you have been amassing gold from your Seafaring trait since round one, and used your early contacts from the same trait to trade technologies for even more gold whenever possbible - and, ideally, you didn't spend a single piece of it during the game so far. That is a pretty treasury you have there. Also, you will of course shut down all research after you reach Invention. You will not need it for the next 20-50 turns, as you will get techs from desperate peace treaties from here on. That gives you the possibility to upgrade a sizable Berserk army fairly quickly.
And then you go forth and shred. It's that simple. Or rather, it can be more complex and even stronger, if you consider some additional perks. I already mentioned several times that Berserk replaces Longbowman, and why this is immense fun. But consider the implications fully: you still have Knights! That might be a strange and trivial realisation, but remember that most other awesome Medieval Age UUs replace Knight. Never forget that you can still build it with Vikings! Adding some Knights to the Berserk mix for specialised missions can augment your already formidable horde to absolute invincibility. Unit mix is key! When playing extremely UU focused, that is often forgotten - don't forget it with the Vikings! And the cherry on top of all of this is of course that you have the Militaristic trait - resulting in promotions and Military Leaders galore. The Military Leaders are quite wasted on Berserk, since the Armies won't fit in the ships you have available, but your Knights will thank you (seriously, do not forget about building some of them Knights!). Another, minor, accelerator of your play is that you have a good shot at conquering Knight's Templar from someone - again, something to put into your armies.
It is occasionally leveled agains the Berserk that it doesn't upgrade. To which I reply: why would you ever want to upgrade this beast!? You would only upgrade away its greatest feature - the amphibious assault. Berserk have a high enough attack to stay a central component of every Viking army right up until Marines come along (if you fancy to go for Amphibious Assault at all).
This is, of course, a rather one-minded playstyle. But as I pointed out, you can try to go for something more peace-oriented, and how much conquering you do is ultimately up to you. Furthermore, all this will require precise timing, but nothing approaching witchcraft. If you still turn your nose up at the Vikings, and say that they might prosper with a decent starting point and favourable map, but won't be able to salvage a crappy start, like the Maya or other builders might be able to, let me suggest to play with settings such as: huge world, 3 billion years old, 80% water, Archipelago, cold, dry, raging barbarians, 16 Civs - the most inhospitable surroundings you might imagine. Play these settings with Vikings. Don't be discouraged by most starts you draw. Only tundra, desert, and mountains in the first screen is probably something to throw, plains are awesome, grassland with bonus resources is probably too good to even have fun with the game. Seriously: your starts will look like crap by any other standard, but just try. The Vikings will dominate such maps like none other. The playstyle I just described is extremely self-sufficient (did I mention that Berserk do not even have a resource requirement?).
You can probably salvage a significantly higher proportion of bad starts with the Vikings than with any other Civ. You can turn a few towns clinging to the cliffs of a barren wasteland into an empire spanning the world withing half an Age. Get into the Viking mindset and use it! I was never disappointed (even if the game ultimately crumbles, it is usually tremendous fun to play).