Henry and Richard are foils for each other. Where Richard is passive, Henry is active. If Richard relies too much on language, Henry rarely reveals his thoughts and opinions. He also prefers to let his bold actions speak for themselves. Both men are grandsons of Edward III, and both use that fact to defend their claims to certain rights (like titles and land). Also, Richard believes God will take care of him and spends much of the play reflecting on his situation and his identity. Henry is not an introspective man, and he doesn't wait around for God to make things happen for him.
There are two important father-son relationships in the play: Gaunt's with Henry Bolingbroke, and York's with Aumerle. However, they work out very differently. Whereas Gaunt stands by his son and encourages him to flourish despite the king's decision to banish him, York turns viciously on Aumerle when it turns out Aumerle has been part of a plot against the new King Henry. FYI, father-son relationships become an even bigger deal in the next play. Shakespeare hints at this when Henry Bolingbroke complains that his son, Prince Hal, is a loser.