Shall it for shame be spoken in these days, 
Or fill up chronicles in time to come, 
That men of your nobility and power 
Did gage them both in an unjust behalf 
(As both of you, God pardon it, have done) 
To put down Richard, that sweet lovely rose, 
And plant this thorn, this canker, Bolingbroke? 
(Act 1, scene 3, lines 174–80)

Hotspur addresses these lines to Northumberland and Worcester on the subject of the Percy family’s role in dethroning Richard II and getting Henry installed as king. In rehearsing this history, Hotspur implicitly calls into question the legitimacy of Henry’s rule. Although the Percys supported Henry in his plan to dethrone Richard, they claim to have done so under the assumption that the transfer of power would be bloodless. Thus, when Richard was assassinated not long after being deposed, the Percys felt betrayed. Hotspur’s speech here reflects the Percy family’s regret about helping to replace “that sweet lovely rose” with “this thorn, this canker.” Henry doesn’t deserve the crown.

The skipping king, he ambled up and down 
With shallow jesters and rash bavin wits, 
Soon kindled and soon burnt; carded his state, 
Mingled his royalty with cap’ring fools, 
Had his great name profanèd with their scorns, 
And gave his countenance, against his name, 
To laugh at gibing boys and stand the push 
Of every beardless vain comparative; 
Grew a companion to the common streets, 
Enfeoffed himself to popularity, 
That, being daily swallowed by men’s eyes, 
They surfeited with honey and began 
To loathe the taste of sweetness, whereof a little 
More than a little is by much too much. 
(Act 3, scene 2, lines 62–75)

In these lines, King Henry describes his predecessor, Richard II, in unfavorable terms. This passage comes from a longer speech, in which Henry compares his own conduct with that of Richard’s. Whereas Henry gained the public’s favor by keeping himself scarce and only showing his face in court at key moments, Richard was unscrupulous in his dealings. He caroused openly with anyone and everyone, “mingl[ing] his royalty with cap’ring fools.” The court quickly tired of Richard’s behavior, which Henry strongly implies was dishonorable for a king, and thus undermined the legitimacy of his rule. The king must stand above, not among, his subjects.

Douglas: Another King! They grow like Hydra’s heads.— 
I am the Douglas, fatal to all those 
That wear those colors on them. What art thou 
That counterfeit’st the person of a king? 
King Henry: The King himself, who, Douglas, grieves at heart, 
So many of his shadows thou hast met 
And not the very king. 
(Act 5, scene 4, lines 25–31)

During the battle that takes place in act 5, the Earl of Douglas learns that there are several “counterfeit” kings on the battlefield. These figures dress like the king to draw attention away from the real king. The first counterfeit Douglas kills is Sir Walter Blunt, and he evidently goes on to kill several more. This explains why, when he finally comes upon the real Henry, Douglas likens the king to the mythical “Hydra,” which grows two heads for every one that’s cut off. Unsure of whether Henry is another fake, Douglas asks, “What art thou / That counterfeit’st the person of a king?” His assumption that Henry is just another fake symbolically reflects the play’s larger question about the legitimacy of his rule.