Speech and Dialogue
Since this is a play and not a novel, where a narrator can describe things, one of the best ways to understand a character is by paying attention to what he says and how he says it. Richard, for example, spends a lot of time talking – mostly about himself and what it means to be king. Henry Bolingbroke, on the other hand, doesn't talk about himself much at all, and sometimes he literally has nothing to say. This tells us that he's much less interested in talking than in action – like kicking the king off the throne.
When Henry Bolingbroke does talk, he tends to speak in blunt English, which tells us that he's direct and straightforward. Richard, on the other hand, tends to use flowery language and is a big fan of metaphors and imagery. This suggests that he's a bit of a drama queen and that his words can't always be trusted. For example, when Richard gets stripped of his crown, he makes a big speech about how he's taking "this heavy weight [the crown] from off [his] head" (4.1.5). Bolingbroke doesn't make any fancy speeches – he just says "Are you contented to resign the crown?" (4.1.12). Translation: "Hurry up and take off the crown already. We don't have all day."
The way a character responds in a crisis can tell us a lot about what that person is made of. Take Richard II, for instance. Richard gets himself into lot of trouble because he never really does anything to defend himself or his right to the crown. When he finds out that he can't defend himself against Henry's big old army, he sits down on the ground (literally) and feels sorry for himself (3.2.9).
Henry, on the other hand, is all about action. When he feels like justice hasn't been served, he does something about it: he challenges Mowbray to a trial by combat (1.1), amasses a HUGE army to confront Richard (2.3), and helps himself to the crown when he sees an opportunity (3.3). What does Richard do when he's down and out? He boo-hoos about how unfair it all is. Shakespeare seems to be telling us that if you want to be a strong king, you've got to have some guts and be willing to flex your muscles when necessary.
Thoughts and Opinions
There are two ways characters think about kingship in this play, and their attitudes toward this issue help us understand their actions and what makes them tick. On the one hand, some characters (especially Richard) think Richard has a right to be king, because 1) he's inherited the throne, and 2) a king's right to rule comes from God. According to this view, since Richard has been divinely appointed, he doesn't have to answer to anyone except God. If the king's subjects have a problem with his style of rule, that's just too bad. Plus, if anyone questions the king's authority, they're basically questioning God. This is why Richard acts like such a jerk at the beginning of the play and why a lot of the nobles in his court let him get away with Gloucester's murder.
On the other hand, some characters (especially Henry) think a king's power depends on his subjects and whether or not they think he's doing a good job. Characters who think this way are more likely to challenge Richard's authority instead of kiss up to him – like Henry, who raises an army and kicks Richard out of office.