By the end of the play, King Henry has pretty much checked off everything on his short "To Do" list. Recall that his "To Do" list looked something like this:
1. Demolish the French army: Check. Henry kicks serious butt at the Battle of Agincourt (4.3-4.7), which is nothing short of a miracle because Henry's army was totally outnumbered but somehow managed to pummel the French anyway.
2. Become the King of France: Check – sort of. Henry doesn't get to rule France right away. Still, he forces King Charles VI to name the "heir apparent" (next guy in line after Charles dies) to the French throne. As a bonus, Charles also throws in his daughter, Catherine, so the couple can get hitched and have a son that will also rule France one day.
Wow. That's pretty impressive, don't you think? At the play's end, though, Shakespeare doesn't exactly roll out a banner that says "Mission Accomplished." Instead, he sends out the Chorus to deliver one seriously depressing Epilogue reminding us that, after all the bloodshed and struggle, things don't actually work out very well for Henry and his family. Check it out:
Small time, but in that small most greatly lived
This star of England: Fortune made his sword;
By which the world's best garden be achieved,
And of it left his son imperial lord.
Henry the Sixth, in infant bands crown'd King
Of France and England, did this king succeed;
Whose state so many had the managing,
That they lost France and made his England bleed:
Which oft our stage hath shown; and, for their sake,
In your fair minds let this acceptance take. (Epilogue)
Translation: "We've just showed you (the audience) what an awesome warrior King Henry V was, but we should probably remind you that our portrayal of Henry's success at Agincourt is just a brief, flashing glimpse ('small time') into Henry's life. Catherine and Henry V will go on to have a son, but Henry VI will eventually screw things up big time by losing France. Psst. Shakespeare wrote all about this in Henry VI, Part 1 and Henry VI Part 2 so you should totally go and see those plays if you haven't already."
Brain Snack: The historical Henry V ended up dying of dysentery (basically a really bad case of diarrhea) in 1422 after going back to France for yet another military campaign.