Did they ever confirm that he was schizophrenic? I didn't know that. But here's my two cents.

Pretend you're Harvey. You're trying to clean up your city the best way you know how. You're dating Rachel, who shares your vision of a better Gotham. You've also got Joker running around, blowing stuff up. Batman is trying to stop him, but you're conflicted about him, especially since, up to this point, he's failed. Then, BOOM. The Joker blows up Rachel, and Batman saved you instead of her. And, you're brutally disfigured in the whole deal. You wake up, and all you know is that Rachel is dead because Batman failed to stop the Joker, your system failed to stop him, and somehow, you're still alive to see the end result.

Remember his quote: "You either die a hero, or you live long enough to see yourself become the villain." Maybe some part of him knew that, if he died and Rachel lived, he would be remembered as a hero. And not only was that stolen from him, he's ugly now. Maybe he isn't the villain yet, but he looks the part. And life isn't fair. It's chance. So why not use chance to decide your actions?

Absolutely. Harvey’s history of mental illness was paramount to him becoming Two-Face and his mental breakdown.

What I think is Two-Face was just to get back at Batman but, really the whole situation is just really hard to comprehend without more background information on Two-Face in the Dark Knight. So, vengeance is what I'm guessing.

Wow