Why obsess over Alucard? The short answer: because he's so darn cool.
The slightly longer answer: because he is the embodiment of Hellsing. The mood of the series is his - the cockiness, the violence, the darkness, the style, the humor in some places and the deadly seriousness in others. If you like Hellsing, you can't help but like Alucard.
(The same thing is, I think, found in Shine. It takes a lighter tone on the world of Hellsing and life in general, and its Alucard does less shooting and more mocking.)
The extended answer:
The guy has style. His long and uniformly red ensemble is very classy; his motions are elegant and minimalist (and it doesn't hurt that he never has to worry about dodging bullets); he wields his guns with the kind of artistry that comes when you're doing art for art's sake.
He does everything with superb confidence. Just as you can only be a wiseguy if you're always right, you can only be this overconfident if you actually are as awesome as you think you are. Alucard is that good. And he knows it.
Alucard's presence makes otherwise bloody, ugly fights look classy. The elaborate battle scenes that crisscross the manga are usually his doing, and if not they're his style. Swirls of torn fabric and bodies, swarms of bats, whorls of darkness, pools and splashes of inked-in blood - this is what a fight with Alucard looks like. It's striking enough to distract you from the actual body count.
The philosophy of the series usually comes from Alucard. It's his morbid wisdom that gives rise to the memorable one-liners ("Giving in is what kills people. When you refuse to give in, you transcend your humanity"). The psychology behind this, too, is fascinating. Immortality does strange things to the brain.
His fangirls say he's sexy. Odd as it may seem, he does have his bishōnen moments. But you needn't find him hot to appreciate that he looks cool.
And that brings us back to the short answer. There's no getting around it. Alucard is obsessed over because Alucard is, simply and undeniably, cool.