Why does Chillingworth want to exact revenge on Mr. Dimmesdale in The Scarlet Letter by Nathaniel Hawthorne?
This is a tricky question to answer for two reasons. First of all, I'm not sure how far you are in the reading so I don't really know how much you know about these two men; second, I'm not sure I would call what Chillingworth wants "revenge." Forgive me if I reveal too much.
In chapter four of The Scarlet Letter by Nathaniel Hawthorne, We learn that Roger Chillingworth is actually Hester Prynne's long-lost husband (which means his name is actually Prynne). He has been gone for two years, and he arrives back home to discover his wife standing in a shameful public display on the scaffold--with a baby.
Certainly Chillingworth has every reason to exact revenge on both Hester and her lover, but that is not what he says when he meets with her in her prison cell. He says this to Hester:
"We have wronged each other.... Therefore, ...I seek no vengeance, plot no evil against thee. Between thee and me, the scale hangs fairly balanced. But, Hester, the man lives who has wronged us both! Who is he?"
Clearly he is willing to take at least half the blame for this shameful scenario, and he has no intention of exacting revenge on Hester. He does, however (and not surprisingly), want to know who her partner in this crime/sin is. Hester, of course, will not tell him, and Chillingworth continues.
"He bears no letter of infamy wrought into his garment, as thou dost; but I shall read it on his heart. Yet fear not for him! Think not that I shall interfere with Heaven's own method of retribution or, to my own loss, betray him to the gripe of human law. Neither do thou imagine that I shall contrive aught against his life; no, nor against his fame; if, as I judge, he be a man of fair repute. Let him live! Let him hide himself in outward honor, if he may! Not the less he shall be mine!"
Chillingworth says he does not intend to seek revenge against the man or harm him in any way, nor does he intend to get in the way of God's punishment of the man, who is undoubtedly suffering from significant guilt. Though Chillingworth wants to know who the man is, he has no explicit plans to exact revenge on him. Quite the contrary, in fact. Despite his assurances, though, we are not convinced that Chillingworth intends no harm, and neither is Hester.
The reason this information is essential to your question, of course, is that Arthur Dimmesdale is Hester's illicit lover.
In his conversation with Hester in her jail cell, Chillingworth says, several times, of Hester's co-sinner, who turns out to be the Reverend Arthur Dimmesdale, "He shall be mine!" or "Thou wilt not reveal his name? Not the less he is mine." He says that he will not "interfere with Heaven's own method of retribution, or . . . betray him to the gripe of human law." Therefore, Chillingworth vows that he will not turn the father of Hester's baby over to the town authorities but that, instead, this man will "be his." What can this mean? Hester is clearly concerned and interprets these words as very threatening. She says, "Thy acts are like mercy . . . but thy words interpret thee as a terror!" She means that what he says he will do sounds merciful—that he will not turn the man over to the law—but the words themselves are frightening. Chillingworth does want revenge because, as he says to Hester, this man "has wronged us both!" He feels that he shares some responsibility for Hester's current condition, and so there is a balance of blame between the two of them; not so for the man she slept with, Dimmesdale. However, he makes it clear that his revenge will not take the typical or, rather, expected course.