Literary critics refer to Henry V as a "history play," a genre that portrays English historical events (well, history according to Shakespeare) that resonate with current political issues, including matters of kingship, constitution, and rebellion. Yep, that's a mouthful all right. Let's break it down and talk specifics.
Portraying English historical events: Check. Henry V covers historical events immediately before and after the Battle of Agincourt (1415), a major turning point in the Hundred Years War, which was waged between the French and English over the French crown.
Historical events resonate with current political issues, including matters of kingship, constitution, and rebellion: Check. (When we say "current" political issues, we mean around the 1590s, when the Henry plays were written.) As Shakespeare was writing about Henry's miraculous victory at the Battle of Agincourt, England was facing some problems in the foreign affairs department – namely, a long, drawn-out war with Spain and the Earl of Tyrone's Rebellion in Ireland. We also want to say that Shakespeare's portrayal of the problem of kingly succession (in both France and England) echoes a major concern in Elizabethan England. At the time the play was written, Queen Elizabeth I was in her 60s and had no heir to inherit the English throne. As Shakespeare's original audience watched events unfold in the play, Henry V's struggles with the French would have resonated with the problems Queen Elizabeth faced.
Spicing up history with some fiction: Check. Like we've said elsewhere, the Henry plays aren't just all about history. Shakespeare also dreams up a few rowdy fictional characters (like Pistol, Nim, Bardolph, and Mistress Quickly) in order to inject some comic relief into an otherwise serious play.