I do not disagree with what has already been said, but I think that the original state of Chillingworth's "heart" is not as bent on revenge as it later becomes. Admittedly, he starts out to find out who the father is, and he has been hurt by Hester's behavior --- although he realizes that part of it is his fault. One of the things the story is about, however, is how the obsession with revenge corrupts the one seeking it to where Hester can say, "That man's sin is blacker than our own --- he has violated in cold blood the sanctity of the human soul." From seeking to reveal the father, perhaps for revenge, perhaps to make him own up to what he has done, Chillingworth becomes the most evil (?) character in the book ... but it is the transformation that is interesting, not the evil.
Chillingworth intends to hurt the father of Pearl, not by physically harming him, but by emotionally punishing him. This is exactly what he does when he suspects Dimmesdale of being the father of Pearl. He goads him and questions him with hypothetical questions designed to make Dimmesdale very uncomforable. His plan succeeds because when Chillingworth takes a look at Dimmesdale's chest while Dimmesdale sleeps, he sees an "A" above the heart of the minister. This was probably carved or marked by Dimmesdale himself as a result of his own guilt at having fathered Pearl and then not confessing the deed publicly.