When Shakespeare wrote the history plays, he was exploring power, politics, and generally what makes a good ruler. Shakespeare used England's recent and turbulent history to explore these questions.
The evidence of power politics is readily evident from the opening scene of the play. Henry is the king but since he came to the throne by usurping Richard II, he didn't feel secure. Was Richard a good king? By all accounts, no. Was Henry justified in returning to England to claim his father's land and title? Yes but this did not give him the right to take the throne from Richard. In scene one, he finds that his former allies have turned in rebellion against him. The guilt ridden king then responses with an iron fist once the treacherous Worcester misrepresented the king's terms.
On the home front, the young prince, Hal prefers the company of Falstaff and his cronies to the court. He is not happy with the idea that one day he will be king. He refers to it as a debt he didn't incur but must pay regardless. It would appear that the young prince is a wastrel since Falstaff is definitely not a good role model. He also knows that one day he will have to reject the old rascal. One could say the young prince was sowing his wild oats but he was also down among the people that he knows he will one day rule. What better way to get to know your subjects?
Hotspur would seem to represent the ideal. He seems to embody all those qualities of a great leader which are seemingly lacking in the young prince. (Here Shakespeare has manipulate history for dramatic purposes since the real Hotspur was considerably older than the young prince.) Hotspur was aptly named since his flaw was his hot temper and this lead to his death.
It would seem that Shakespeare has given us several examples of power politics. Bolingbroke used power to gain the throne unlawfully but was not very effective playing politics with his nobles which lead to a rebellion. Hotspur gave into his emotions rather than play the game and died as a result. Prince Hal got to know his subjects on a personal level and when he became king, he had learned his lessons well.