In what way does Mr. Darcy's opinion of Lizzy differ from what she claims to be her character and how does he see himself?
Since Pride and Prejudice is told from a third-person subjective point of view, the only private thoughts we're privy to are Lizzy's. Thus, we must base our opinions of other characters on their actions (as reported by Lizzy and as reported by the narrator) and their reactions to our protagonist, Lizzy.
If you examine Mr. Darcy's reactions to Lizzy and her family, it becomes clear that his worldview is heavily influenced by classism. A member of the upper class, Darcy studies Lizzy's family and sees poor breeding, bad manners, multiple examples of bourgeois behavior. This gives him the impression that Lizzy is a "diamond in the rough" -- that is, a girl who, despite her breeding and limited social standing, has managed to attract his affections. The fact that she does so disturbs Darcy. This differs from Lizzy's worldview in a few very important ways. Lizzy, although aware of her social standing, doesn't let it define who she might be or what she might do (for instance, unlike her mother, she refuses to obsess on the idea of "marrying up). She understands her family's poor behavior reflects poorly on them, but she doesn't view it as a class issue.
Darcy is keenly aware of his own social standing, and he purposely separates himself from others He doesn't realize how arrogant and cold he seems. Darcy has never had to question his world view or think about how others view him; as a powerful man, his opinions and attitudes have rarely been challenged.