Winchester starts off the play with a complaint and the threat of a power play, and he pretty much continues that way. Dude's got quite the chip on his shoulder because nobody gave him a job in Act 1—in fact, he's so annoyed that he threatens to kidnap the young king and use him in a bid for power. We never hear any more about this, so presumably he drops it, but he does spend the rest of the play jockeying for power. It's kind of his thing.
Winchester tries to lock Gloucester out of the Tower, which is a bit like trying to lock Nick Fury out of SHIELD: pretty mean and just generally a bad idea. He and Gloucester argue the whole way through the play, and sometimes their men get in on the action, too. It's so bad that the Mayor of London even has to stop them—twice.
Adding to his sleaze factor, Winchester also seems to have bribed the Pope to make him a cardinal. You're not really supposed to get on in the clergy by paying bribes… But then again, Winchester doesn't seem to be in it for spiritual reasons. Nope—he's all excited because he thinks being a cardinal will let him push Gloucester around.
Winchester's not very sympathetic as far as characters do, but maybe he has a reason to be hoping for status: It sounds like someone in his family tree was born outside of marriage, which would make it harder to inherit wealth and power. And he does get left out of the important jobs in Act 1. So Winchester may feel like he has to fight harder than the other nobles to get somewhere in life. It sure sounds like he's been trying to get some respect for a long time—and like he's in it for the long haul.