Shakespeare's history plays are obsessed with royal power, especially the question of who has a right to rule and why. Should the throne be inherited by an eldest son? Can anyone just come along and take it by force if they feel like it? In this particular play, the English King Henry V makes a sketchy claim to the French throne and goes to war in order to secure his position as France's next king. Meanwhile, his claim to the English throne is being called into question by those who think he's doesn't have a legal claim to the crown. (After all, Henry only inherited it after his dad stole it away from Richard II.) In Henry V, Shakespeare also considers what it is that makes a good king and admits that, sometimes, being a successful monarch often involves being a not-so-nice person.
Henry V's claim to the French throne is dubious at best. Henry claims that he has a right to rule France through his grandmother's family tree, but, if the exact same argument were used to decide who should rule England, Henry wouldn't be the king.
The play suggests that being a successful king involves a willingness to be ruthless.