Prince Hal has a target on his back in The First Part of Henry the Fourth, a target placed there by himself, and one that everyone can see.
In the opening scene, when Westmoreland comments that Hotspur’s exploits are those “for a prince to boast of” (I.i.77), King Henry IV can only lament the “riot and dishonor stain(ing) the brow // Of (his) young Harry” (I.i.85-86). The king sees it, and the prince does, too, as he relates in his soliloquy at the end of Act One, Scene Two, where he references his own “loose behavior” (I.ii.201) and his “offense” (I.ii.209), but promises a future “reformation” (I.ii.206). It’s all in his master plan, he says, to be the bad boy now and “redeeming” (I.ii.210) himself later.
Until then, however, he will be looked upon as dissolute. Hotspur, when contemplating a way to kill him, doesn’t devise a noble death on the battlefield or even poisoning the “nimble-footed madcap Prince of Wales” (IV.i.95) with wine, but with “a pot of ale” (I.iii.232). His drinking partner (and surrogate father figure) Falstaff knows the reputation as well, as he tell the prince (while play-acting the role of the king),
Shall the son of England prove a thief and take purses? a question to be asked. There is a thing, Harry, which thou hast often heard of and it is known to many in our land by the name of pitch: this pitch, as ancient writers do report, doth defile; so doth the company thou keepest
— II.iv.396-401
Falstaff sees the problem in the company Hal keeps, sees the problem in himself.
The true father sees the problem as well, though his guilty conscience seems to put some of the problem on his own shoulders for “the hot vengeance and the rod of heaven // To punish (his) mistreadings” (III.ii.10-11).
This dissolute reputation all changes, however, when Hal makes his challenge to Hotspur (in absentia) for a single fight:
For my part, I may speak it to my shame,
I have a truant been to chivalry;
And so I hear he doth account me too;
Yet this before my father's majesty--
I am content that he shall take the odds
Of his great name and estimation,
And will, to save the blood on either side,
Try fortune with him in a single fight.
— V.i.93-100
He realizes, with “shame,” his truancy to chivalry and his high office. But he goes beyond admitting his problem (as they say that’s the first step), to actually trying to make it right. He doesn’t merely agree to go to war, but he is willing to take on the “active-valiant” (V.i.90) Hotspur in a single fight to “save blood” not just on his side, but on both sides. The prince is “redeeming” himself as he speaks.
The reformation is complete when others note the change as well, as when Vernon tells Hotspur of the challenge. Hal, Vernon says, has become “like a prince indeed” (V.ii.60), taking time to make “a blushing cital” (V.ii.61), or account, of himself by “chid(ing) his truant youth” (V.ii.62). Vernon is impressed:
let me tell the world,
If he outlive the envy of this day,
England did never owe so sweet a hope,
So much misconstrued in his wantonness.
— V.ii.65-68
He has now proved himself to friend and foe alike, and only now is able to defeat his “foil” (I.ii.208), Hotspur with actions on the battlefield.