I would reject the notion of a "Vampire" in Dark Sun; vampires are too horror/fantasy whereas Dark Sun is more metaphysical and science-fiction based. I'd rather go with a Kashiga (sp).
Kaisharga. I'd go with a t'liz as the base creature, personally, though
.What level are we talking about here, anyway?
If you feel like it, you could make a custom undead that has some specific vampire features but not others. Traditionally, each undead in Dark Sun is unique, within certain general types. So you could make an Athas-specific undead. Vulnerability to the sun need not necessarily be part of it. If you have access to earlier-edition Dark Sun material, check out the Terrors beyond Tyr monster manual and Dragon #173's "A Letter from the Wanderer" article (both for 2e) and Terrors of the Dead Lands for 3e - they've all got some great treatments of DS undead with suggestions for differing characteristics that you could use to make a unique Athasian vampire.
I would reject the notion of a "Vampire" in Dark Sun; vampires are too horror/fantasy whereas Dark Sun is more metaphysical and science-fiction based. I'd rather go with a Kashiga (sp).
Kaisharga. I'd go with a t'liz as the base creature, personally, though
What level are we talking about here, anyway?
The question at hand is for a player to use the Vampire class. The Kaisharga and T'liz really aren't options. Unless you are simply going to refluff the Vampire, which takes care of the problem in the middle quote, since you can easily refluff the "horror / fantasy vampire" into Dark Sun's Kaisharga. That is how the Kaisharga came about in the first place.
For the OP:
If you look at some of the older art, walking about Athas completely covered is not all that uncommon. A wise desert traveller covers their head to prevent overheating, and covers their eyes and mouth to protect them from sand.
The OP's player might be interested in a vampire's powers and abilities as much as any other facet of the creature, so suggesting Athasian undead is totally an option. Don't you trample on my geekish enthusiasm, man!!! 
The cool thing about Athasian undead is that they are meant to be custom-built. So t'lizes are described as appearing "much as they did in life", although with extremely pale skin. "They must also feast upon the life energy of the living to sustain their own energies." And "They cast no reflection or shadow and they move in complete silence."
So right there you have an Athasian undead that was clearly inspired by the vampire (not the kaisharga, which, as everyone knows, was based on the lich, pfft!) The OP can reskin the vampire and call it a t'liz, or build a t'liz that's like a vampire or something. Or just use the vampire in a big robe and a nice hat or something .
I find it very awesome to be a vampire that is constantly wrapped up like a mummy to protect itself from the sun.
I'm of the same opinion. It has a 'Blade 2" flavour to it. The idea is not at odds with playing in a setting where the sun is a killer for the normal population. A vampire just has a greater risk. It also has the possibility of being rather comical if NPC's try ripping the clothing off.
Think about what happens when they get cut by a knife or set on fire.
Well that's the risk you take playing a Vampire. Easy come and easy explode/ash/wither/age. Should be fun to describe.
The second is where you could try giving a nice Athas spin to the whole feel of the vampire. The idea of a vampire is something that feeds on other people, draining sustenance to supply what they lack. For the vampire in traditional lore they need blood to (un)live, blood to keep their body going.
For Athas the first thought is moisture as an alternate need, after all water here is more precious than blood (except defiling, you evil, evil arcane casters). But that seems really easy to me. If you want to give your vampires a nice spin, how about having them craving and draining not just life-force, but something more potent?
How about vampires feeding not just on the blood, but the "faith" of people. Hear me out. Why would vampires spawn on Athas? Why would something that finds the sun Anathema spawn here? In the history of Athas, there were once Gods but they left. And what happened to the people that once believed in them, what happened to their power, their needs? You're a cleric and then your gods loose, get destroyed or flee. You've become accustomed to divine power and now it's gone. So you wander for a time, hunger setting in, a need for power, for divine power, for faith, for something that you lack. Your God is gone and it's left a gaping hole inside you. And you can only sustain this need by drawing on others. You lack faith yourself, but all these people that believe in Sorceror-Kings or the raw elements or the spirits. You can feed, you can fill that hunger.
The way I'd spin and open up some options for vampires is have them hungering for faith, hungering for the belief that others have that you lack. The curse of vampirism is a lust not just for blood, but for the faith that other's possess that you supremely lack yourself. The original vampires were former divinely empowered people that then passed on their curse. Prowling the world looking for sustenance, blood and faith.
We've got a feat in Dragon 400 (The Hybrid Multiclass article, PG 5, Divine Vampire) that gives us some wonderful options. It takes away the vulnerability to sunlight and radiant and gives context to the vampire. Yes you're looking at a hybrid class progression, but it lets you create something much more interesting. It also gives your player options to play divine classes (if you so wish). It adds a really nice spin to it all, a divine caster who needs to feed on others to power their abilities. If you want to continue to be this potent divine character, you need to keep feeding on others, living on their blood and empowering you own divine abilities (which would be strange and unusual to others) through their faith.
I dunno, just something wierd, quirky and hopefully quite Athasian. Hope it helps.