In Claudius's soliloquy, we learn that Claudius did actually kill his brother. In his soliloquy, Claudius talks about his crime, and he says that it "bears the mark of Cain" who also killed his brother. Although we had been suspicious of Claudius throughout the story, this is the moment when our doubts are confirmed. We also learn that Claudius is extremely guilty about his actions, but that he believes that time and God's mercy will eventually help him overcome this feeling of guilt.
Yes he does, he does not give up on what he believes in. In his case he believes in equality of all. A resilient person can be described as someone who has a lot of perseverance.