Take a story's temperature by studying its tone. Is it hopeful? Cynical? Snarky? Playful?
Somber, Patriotic, Ironic
Henry V is a war play, so when we ask about its tone (the play or author's attitude toward its subject matter), what we're really asking is this: What's the play's attitude toward war? Specifically, what's the play's attitude toward Henry's decision to invade France?
The answers aren't cut and dried in this play because the tone shifts between patriotic fervor for Henry's campaign against the French and it's recognition of the horrors of warfare. On the one hand, the play celebrates Henry's triumph over the France, which seems miraculous given that the English troops were exhausted, sick, and seriously outnumbered at Agincourt. The play is also chock-full of patriotic speeches that suggest warfare is patriotic and ennobling. (The clearest example of this is Henry's famous St. Crispin's Day speech, where he insists that the men who fight alongside him will become his "band of brothers.")
On the other hand, Shakespeare goes out of his way to show us the horrors of warfare, which involve brutal hand to hand combat, raping, pillaging, and endless suffering. As Exeter points out, neither side can escape "the widows' tears, the orphans' cries, / The dead men's blood, the pining maidens' groans, / For husbands, fathers, and betrothed lovers" (2.4.2).
We talk about this at length in "Quotes: Warfare," so check it out if you want to think about this some more.