How does Roger Chillingworth change during the course of the novel?
Even early in the novel, as Hester stands on the scaffold with her infant daughter, there are signs that Chillingworth has the capacity, at least, to be evil. The narrator says that, when he sees her upon the scaffold, "A writhing horror twisted itself across his features, like a snake gliding swiftly over them." Snakes, especially in texts that comment on Christianity (Puritanism, specifically, in this case), are often symbolic of evilness as a result of their connection to the Devil in the story of the Garden of Eden. Furthermore, Hester is clearly afraid to be alone with Chillingworth. The narrator describes her thoughts to us: "Dreadful as it was, she was conscious of a shelter in a presence of these thousand witnesses. It was better to stand thus, with so many betwixt him and her, than to greet him, face to face, they two alone." This would seem to indicate that Chillingworth is a man to be feared.
By the time that Chillingworth becomes the physician of Reverend Dimmesdale, many people in Boston begin to "[affirm] that Roger Chillingworth's aspect had undergone a remarkable change [....]. Now, there was something ugly and evil in his face, which they had not previously noticed, and which grew still the more obvious to sight the oftener they looked upon him." As Chillingworth becomes more and more consumed by his desire to exact revenge on Hester's co-sinner, whatever good qualities he once had seem to fall away as he is given over to moral darkness. Even his face appears to be physically darkening.
Later, after Chillingworth has verified that Dimmesdale is Pearl's father, Hester tells him that she pities him, "'for the hatred that has transformed a wise and just man to a fiend!'" Hester was well-acquainted with Chillingworth's character prior to the events that take place in the text, and so her words are perhaps the most reliable when it comes to describing how her husband has changed.
At the beginning of the novel, Chillingworth is an intelligent older man who has been held captive by the Indians for over a year. When he sees Hester and finds out what she has done, he is shocked, angry, as well as regretful for what he feels is his part in causing the affair. He seems to be a rational, reasonable older man. After meeting with Hester however, he vows to discover and take revenge on the man who had an affair with Hester. As he continues to seek revenge against Dimmesdale, the novel says he becomes more and more evil looking. His entire life revolves around this revenge, and he becomes bound by it. The book even says that at times he looks like the very devil. His appearance becomes more ugly and misshapen. By the end of the novel, when Dimmesdale confesses, Chillingworth has no option except to die, because his whole life has been centered around revenge, and the revenge is no longer possible.
Another interesting addition is that when we last see Chillingworth, in Chapter 24, he has relented, more or less, to become kinder. He has given a substantial inheritance to Pearl, and has become unlike he has been previously, unless you consider his honesty with Hester in the jail earlier in the book to be a hint of goodness.