One of the basic issues the play investigates is how you go about determining who or what deserves your loyalty. Are you loyal to a king? To a country? To your relatives? To the law? To justice? To God? When different systems (like government, religion, and family) compete with each other for your allegiance, how do you choose sides? When Richard starts thinking of himself as above the law, he puts these systems in conflict with each other.
Different characters offer different answers to this question. Aumerle is loyal to Richard. York is loyal to the office of the king, no matter who occupies it. Gaunt is loyal to the kingdom. The Duchess of York is loyal to her son. And the queen is loyal to her husband. So who's right? Shakespeare leaves this for us to decide.
One of Richard's biggest problems is that he thinks he deserves loyalty without having to give it back to his subjects.
Even though Aumerle plots against King Henry IV, he's ultimately more loyal than York, who keeps switching sides.