Heavy is the Head that Wears the Crown
Henry believes that he is rightful heir to the throne of France and thus leads his men into war to take the throne from Charles VI. In so doing, the young King begins his campaign with vicious fire in his belly. And as battles are won we see him begin to contemplate the true price of what he is asking of his men. He wanders the night while his men sleep before battle. He looks like a ghost and his heart weights heavy for the lives of his soldiers who many will lose their lives in the waking hours. Again we see this weight visually when Henry carries young Robin across the field of battle to a wagon full of the dead. Henry is literally carrying the great weight of the war he's waged and the price is that of a child.
Power
Henry V is after a second throne. His desire for Charles VI's crown is more than what he says is his birthright, but more. It's his desire to be the most powerful King alive and if he captures the French throne he will take his place in history as one of the greatest Kings to ever live. But Henry's desire for more comes with a price that his people must pay as they are the ones who will have to fight on behalf of their King so that he may occupy his new dominion. In the scene where Bardolph is hung we watch as the young King has the chance to stop his execution, but if he does he will be weak and lose power amongst his men and thus chooses to allow his friend's death than to save him. All in order to continue to have control of the men he leads towards France.
Class
Henry leads his troops into battle in order that he might take the crown of France from Charles VI. His troops consist mainly of common men and the film makes clear that these commoners pay a great price for their King, as it is they who must wage war on his behalf. It is quite clear the class distinctions when reports come back from the battle on St. Crispin's Day that 10,000 French are dead and 520 English, but the report is clear to state the names of the noble men who dies while those commoners who paid with their lives shall remain nameless in the cause of the King.
