Arthas' flaw is that he ignored the warnings of his trusted friends. First he ignored his childhood friend and his mentor. Then he ignored his uncle Muradin, who warned him about the sword being cursed. Then he usurps his father, a good and wise man who formed the Alliance and defeated the Horde and brought peace to the continent. Who does he listen to? The voice of a sword he was warned about, an evil wizard, and a spider he just met. You have to wonder why he was not well known for foolishness before the game began.
I think Arthas' fatal flaw is that he is primarily ego driven. He was raised to be the perfect prince, not only for his personal qualities but also as a Paladin. And he doesn't take failure well. More importantly, we are slowly weaned into that flaw with each chapter of the main campaign.
When Arthas fights the orcish cultists he doesn't steel himself for the horrors he's about to witness, in stark contrast with Uther who stays calm throught the ordeal. That's not bad in itself. Arthas is young, Uther is old, experienced, and a war veteran. So it's no surprise that Arthas is more emotional, it just seems that he's got a lot to grow as a Paladin.
Arthas starts to show his hand in the undead siege. By that point he'd 'failed' to save the people of Andorhal (though, really it wasn't his fault at all), so when Uther comes in and compliments his defence, Arthas takes that personally as if it was an accusation. You should have done better, you should have been better.
Then two things happen in Stratholme. Arthas takes a drastic measure and puts the city to the sword. We'll never know how many people had actually eaten from the new grain stocks, but we know it hadn't been everyone. The second thing is that Mal'ganis shows up and baits him. From that moment onwards it's not about protecting the people, it's about revenge. That is where Arthas begins to make a series of strategic blunders. He took the enemies' bait because he's emotionally compromised.
This all feeds back into the final cutscene. The sword's whispers are that Arthas 'no longer needs to bear the weight of the Crown'. He can finally live or un-live for himself, with no pretense at morality or responsibility.
Warcraft 3 was no shakespeare but I think it was a well told story with good fundamentals.