Death knights are probably your best, purest option for playing something similar to a vampire character. The entire class is built from the ground up very similarly to vampires, and you can probably do about the same things as a vampire. Let's start from the top.
First, death knights are characters who have already died once. Then, forced from their final rest by Arthas, they are instantly enslaved into a totalitarian, insane society. This is to say, when death knights first come into play, they are part of Arthas's efforts to bolster Acherus. They are only freed from this insane "life" during the Battle for Light's Hope Chapel.
After that exciting opening, death knights are left to wander the world and find their own way. The Knights of the Ebon Blade are obviously focused on the eventual defeat of the Lich King. But you're still left with a character who is essentially undead, possessing significant supernatural power, and an outlook on the world entirely separate from your average character.
Death knights possess more than a few traditionally vampiric abilities. There are not only direct references like Vampiric Blood, but there are also more subtle allusions like Blood Gorged. (If you don't know other RPGs, many vampire systems like Vampire: The Masquerade posit that the more blood a vampire has in its body, the more powerful it becomes.) Powers like Army of the Dead allude to the master vampire's control over other undead.
Ultimately, if you're looking to play a vampire, death knights will probably serve your purposes pretty well. You got the undead angle, the blood angle, and even th ex-slave-now-free angle. There's a lot there to roleplay with, and it does sound an awfully lot like a vampire.
The interesting thing about vampire genre is that it's pretty malleable. You have physical powerhouses from movies like Lost Boys, but you also get the "magic vampire" who is a master of dark arts. Vampire: The Requiem, for example, has an entire host of magic-wielding undead.
If you're interested in exploring something like a sorcery-wielding vampire, I'd recommend the Shadow priest. In terms of WoW lore, the Cult of Forgotten Shadow is a rich opportunity for such roleplay. Created by Forsaken who feel abandoned by the Holy Light, the Cult of Forgotten Shadow is an attempt to understand and deal with the curse of undeath.
Seated in this in-character format, the shadow priest's powers promote the idea of your character being a supernatural undead power house. Shadowform itself is a pretty awesome visual effect, and gives a nice feeling of being a vampire who's about to release his or her immense occult energy. The shadow priest's signature nuke is called Mind Blast, providing a great example of the psychic power of a sorcerous vampire. Vampiric Touch feels like the magical ability of a vampire who has found a way to feed without directly sucking his or her victim's neck.
I'm not usually one to do things like play a Blood Elf but claim it's actually a Darkfallen. However, the in-character history of the San'layn absolutely makes it reasonable that your Sindorei character might know (or have known) one of Lana'thel's brood. Perhaps your Blood Elf was one of the strike force against Arthas, and somehow escaped before the fate of Quel'delar could be known.
Your character could have met Tenris Mirkblood back when he was in Karazhan, and been left with a puzzling obsession about the San'layn ever since.
Of course, if you're a raider who has successfully visited the Crimson Halls in Icecrown Citadel, you could have been permanently affected by what Matt Low calls the "nom-nom mechanic." During the fight against Blood Queen Lana'thel, your character is briefly affected by the Essence of the Blood Queen. Once that effect wears off, your character is overcome by a Frenzied Bloodthirst. That character must bite another raider or become the mind-controlled slave of Lana'thel herself.
While that's just a raid fight dynamic, it's an easy argument to bring it entirely in-character. Presumably, under normal operating in-character raid procedures, everyone is free and hunky dory after defeating the boss. But instead of just assuming your character walks away, perhaps your character has changed in some way. Your character has tasted blood once, and may now wake up at nights, hungering for that brief rush of sanguine power granted by Lana'thel.
If you don't raid normally, a very similar affect is available in Ahn'kahet. In that dungeon, Prince Taldaram randomly uses the ability Embrace of the Vampyr on a party member. When Prince uses this power, he literally pins you to the ground. To most observers, it really does look like he's sucking the target's blood. And since the recipient of Embrace can not be healed during the power, the in-character supposition would be that life-giving magics don't function on something being turned into the undead.
These are obviously just three options. I'm sure people have hundreds of different ways that you could play something like a vampire that still fits within the lore and stories of the World of Warcraft. The goal here isn't to simply drop Edward Cullen in the middle of Silvermoon and LOLRP it up. Instead, pick out the pieces of a vampire-character that you want to portray, and translate that into Azeroth-terms.
The natural caveat here should be that you should try and coordinate this kind of story with your roleplay troupe. Playing a character who's been fundamentally changed by the Blood Queen Lana'thel fight is a great idea. But if you just end up annoying everyone you normally play with, you'll ultimately regret the choice.
I'm definitely interested in hearing how you folks have made vampire-like stories work. I think there's a lot of room for creativity here, and I'd love to be exposed to new ideas.