The Latin expression sub specie Šternitatis is a poetic one, and expresses a lot about vampires, particularly Alucard. It means "from the perspective of eternity."
The key difference between humans and vampires is not the power; it's the mortality. A large chunk of human psychological development hinges on the fact that we know we're going to die sooner or later. Alucard has no such guarantee. (This is why Walter sometimes seems so much more mature.) He hasn't been around forever, but he has forever to look forward to.
So sit back, relax, and make yourself a cup of tea: I'm going to try to explore some of the psychology of Eternity.
The first thing to understand is this: If you stick around long enough, everything will change. Give it a million years, and a mountain will wear away. But humans aren't generally too concerned with mountains. If you want to know what humans care about, read a newspaper: style, politics, trends, events, famous people.
Styles and trends may not last a year. Events are for the moment, and sometimes their impact will last, but often it won't. Look back on politics a few decades ago, and you will find a completely different landscape. As for famous people - the expression "fifteen minutes of fame" is around for a reason.
It takes a million years to move a mountain, but a single century will completely shift everything that matters to humans. Fortunately, that's about as long as the best of us will last. The woman with the longest confirmed lifespan - Jeanne Calment - said, "I've been forgotten by God." She lived 122 years.
Alucard's been around for 567.
You have to shed your humanity if you live for that long. Try to hold on to the cares of a human life, and eventually you'll lose everything. All the people Vlad Dracula ever knew have been dead for most of his existence. The political power he held as a human involved a country that doesn't exist any more.
In this situation, you either stop caring or you go insane.
So Alucard stopped caring. You could argue that it's a defense mechanism. You could also argue that he gained perspective. "Don't sweat the small stuff," the saying goes - the next line being, "it's all small stuff." To a human, it means don't panic about being late or forgetting something or messing something up, because in the big picture your mistakes aren't such a big deal.
For a vampire, the "big picture" is a whole lot bigger. Over the period of half a millennium, what does it really matter if this or that human lives or dies? Do we really care if a car crashes or a bomb blows up or a storm does thirty million dollars' worth of damange? Does it matter if this or that president is elected, when no country has yet lasted for more than a couple of centuries without a revolution?
And when you're planning to last the rest of forever, just about everything begins to look pretty pointless.
Again, this is the kind of stuff that drives people out of their minds. If the inescapable conclusion is that existence has no purpose, then why would you continue to exist? (This is Existentialism 101, by the way.) If you believe in God, you get solace from that - the idea that there's a Planner, and that you're acting part of a greater plan that all works out in the end. (Talk about your "big pictures".) If not, you have to make your own purpose.
Now do you see why Alucard's always looking for a good fight?
Well, there are two reasons, but the first is that it's his self-imposed purpose. It's a quest. It keeps him occupied, gives him something to work towards, imbues his existence with some kind of meaning.
Remember the fight with Luke, how happy Alucard was, and then how much of a letdown it was when he found out that Luke barely had any power at all? The scene was almost identical in the TV series and the manga. If your purpose is to find a good battle, this is one of the most depressing things that can happen.
The other reason is that Alucard wants to be able to die.
He doesn't want to just kill himself; that would be sort of pointless. Rather, he wants to fall in battle with a worthy opponent. It would be an honorable death, and he'd enjoy it. He'd be okay with being killed if he respected his killer.
Alucard was hoping Alexander would rise to the occasion. He didn't. All the smart money is now on Integral. But I digress.
Remember how I talked about everything in Alucard's existence being temporary? There's one exception, and you've probably thought of it by now: other vampires.
The reason that Alucard invests such a large amount of his time in pushing, prodding, and encouraging (in his own way) Seras is that she can now last as long as he can. If a few humans die in a learning experience for Seras, it's perfectly acceptable: those people would be dead in a hundred years anyway, but Seras will still be around. If Alucard's still there, he'll want good company; if not, he'll want his successor to be up to the task.
Seras still thinks like a human. If Alucard prepares her properly, she's in for a radical perspective change. Things look very different when seen sub specie Šternitatis.