We'll tell it to you straight: there's not a whole lot of humor in this one. The tone is pretty somber all the way through, and that makes sense, given what goes down in this play: Henry's nobles argue about everything; he's put through the ringer from both the nobles and the commoners; there's murder and death all over the place, so there's plenty of dark and foreboding moments to go around.
Seriously, folks: strap on your emotional seatbelts for this baby.
On top of that, Shakespeare really messes with our sympathies. Sometimes, we really do end up rooting for the most malicious and evil of the characters. Margaret is bloodthirsty and savage, but her never-give-up attitude and ninja-like guts are something we'll admit we kind of enjoy seeing her play out. Same goes for Cade when he's trying to get the crown: we're repulsed by his brutal actions, but we also admire him for his audacity, his wit, and his guts.
Almost everyone in this play has an angle or a ploy: Eleanor wants to be queen; Beaufort, Margaret, and Suffolk want to get rid of Gloucester; Cade and the commoners want to get rid of all the nobles; and York thinks the crown belongs on his head. With all this scheming going on, it's hard to keep anything straight, and our general code of ethics starts heading into breakdown territory.
We might call this a morally ambiguous tone, or we might just say we're drawn in by all the crazy stuff that goes down in this play. This carries over into the next installment (Henry VI, Part 3) as well.