I've finished the second part of Henry the Fourth and started Henry the Fifth. It didn't feel like much happened in this one, with one shocking exception, when the emissary Westmoreland tricks the rebel leaders into accepting apparently generous terms without any guarantees for their safety; once their armies are dismissed, they are immediately arrested and executed. Because Prince Harry is the hero, you'd think that his allies would be the good guys. Apparently not.
It feels as if the two parts of Henry the Fourth are mostly an elaborate setup for young Henry's transformation. Apparently he's just been tricking everybody into thinking that he's a dissolute, worthless wastrel, and now we get the dramatic fun of watching him come out of disguise and surprise the pants off everybody.