Tetralogy, actually – if you want the whole story, it starts with Richard II and continues through Henry IV, Parts 1 and 2 and Henry V. I like to throw The Merry Wives of Windsor in as well when I teach them, but even I will admit that it’s not an integral part of the saga.
However, Henry V works fine as a standalone – mostly. Here is the Cliff’s Notes version of the bits that are useful to know from the earlier installments.
– In Richard II, Henry Bolingbroke – the father of Henry V – deposes his cousin Richard and becomes King Henry IV. Richard is later murdered in prison, although Henry IV is careful to denounce the murderer in public. (You should imagine Richard’s ghost lurking in the wings the whole time; sooner or later, Bolingbroke’s line is going to pay.)
– Henry IV then spends most of his reign putting down one rebellion after another. On his deathbed, he warns his son to be careful of his fellow nobles and counsels him to “busy giddy minds with foreign quarrels” (advice which Henry V will take to heart).
– After spending his youth hanging out in taverns and playing practical jokes on his drinking buddy Sir John Falstaff, Henry V has a striking reformation after he’s crowned. He dismisses Falstaff and banishes him from his presence. (Whether this is a painful but necessary decision or a truly callous one depends a great deal on how the scene is played.) Falstaff doesn’t appear onstage in Henry V, but his ghost is arguably lurking about, as well. Falstaff’s gang of disreputable friends – Hostess Quickly, Pistol, Nym, Bardolph, and an unnamed boy – are still around. (The king definitely knows who Bardolph is and has been on friendly terms with him in the past; this is important.)
– Early in the play, there is a scene in which Henry V has to deal with three traitors, one of whom is named Richard, Earl of Cambridge. Richard is the brother-in-law of Edmund Mortimer, who is in prison, but whose claim to the throne is in fact stronger than Henry’s. Richard is also the father of the Yorkist line that will eventually depose Henry’s heirs. (Most of this is not made explicit in the play, but everyone in Shakespeare’s audience would have known it.)
Oh, and don’t worry if you have trouble following the opening scene with the archbishops. It’s meant to be about as clear as mud, but the gist of it is that Henry has a claim to the French throne via one of his female ancestors, which the French don’t recognize because their law forbids succession through the female line.