Even though the king's forces enjoyed a victory against the rebels at the battle at Shrewsbury back in Henry IV Part 1, there are still a few dissenters, led by the Archbishop of York, and they're causing a lot of trouble. Also, King Henry hasn't been feeling well lately and he's still worried about what will happen when his son, Prince Hal, inherits the throne.
Even though Prince Hal saved his old man's life at the battle at Shrewsbury back in Henry IV Part 1, King Henry still thinks Hal's a loser, especially since Hal is still hanging out with Ned Poins, a rowdy commoner. In the meantime, Prince John arrests the rebel leaders so the only thing left to worry about is whether or not Prince Hal will reconcile with his father and be a responsible king.
Thinking his father has died in his sleep, Hal makes off with the crown. Whoops, that was not a good call. Things get awkward when the king wakes up from his nap and figures out what's happened. In fact, it seems like all of Henry's worst fears are confirmed by Hal's actions and he accuses his son of wanting him dead so he can be king.
After his father lays into him, Hal delivers a moving speech that finally convinces his father that 1) Hal loves him and 2) Hal wants to be a good king. The reconciliation between father and son happens just in time because shortly after they resolve their differences, King Henry dies (off-stage) and we hear about it from Warwick, who says that Henry has "walked the way of nature" (5.2.3).
Now that King Henry IV is dead, the Lord Chief Justice (the old king's main man and upholder of the law), along with just about everybody else, is pretty stressed out about his future under the new king. That's because, back in the day, the LCJ threw the wild prince in the slammer for misbehaving. Now that the very same prince is the King of England, things could possibly get ugly. Turns out, though, that Hal's committed to being a stand-up monarch. He embraces the LCJ as a "father" and says he's ready to uphold the law of the land.
In one of the play's most painful moments, Hal comes face to face with his old friend, who is an inappropriate companion for the monarch now that Hal has "turned away from [his] former self" (5.5.2). In other words, the king is no longer "wild Prince Hal" and must behave accordingly. When Falstaff (who has lined up on the street to catch a glimpse of the newly crowned King Henry V) eagerly greets his old friend, Hal says "I know thee not, old man. Fall to thy prayers" (5.5.3). Ouch! Hal then banishes Falstaff and orders him to stay at least 10 miles away from him. In doing so, Hal makes a statement to the world that he's a changed man.
Now that Hal's "reformation" is complete, Shakespeare turns his attention to what will happen in the final installment of the tetralogy. Prince John predicts that before the year is over, England will be at war with another nation. (FYI: In Henry V, the newly crowned King of England claims the French throne and invades France.) The end of the play, then, is more of a "to be continued" than a final conclusion.