I don't think, if someone is mistreated as a child they automatically need to become hateful. However a child is treated, in whatever environment, with whatever kind of people they grow up, that is what's normal for them. Harry was probably so used to his situation that the idea the Dursleys should be nicer to him can have crossed his mind only very rarely before he got to know people who treated him better in the Wizarding world. And because he only realized how much better treatment he deserved at the very point when his life became so much better, when he found friends and a real home at Hogwarts, etc., he had no chance to become bitter/ hateful. He did hate the Dursleys to a certain extent during his time at Hogwarts, but he did not project his hate on the whole world because by then he had gotten to know nicer people, too.

Interestingly his upbringing with the Dursleys was good for him in one respect: It made him modest, and thus might have helped him keep a level head and not become conceited when he learned that he was the "boy who lived" and a kind of hero for the Wizards.