What favor does Chillingworth think he is doing for Dimmesdale?
In Chapter 9, Chillingworth becomes the "medical adviser" to Dimmesdale. This is an ironic relationship, of course, because the audience knows (or will soon know) that the cause of Dimmesdale's failing health is in fact, his guilty conscience. Though Chillingworth does offer a variety of medical treatments for Dimmesdale, the truth is that at the same time as he is treating Dimmesdale's symptoms, he is feeding their cause.
Chillingworth and Dimmesdale move in to the same house and spend most of their time together. From the very outset, Chillingworth has correctly suspected Dimmesdale to be the father of Pearl. These two educated men spend most of their time engaged in scholarly debate, Dimmesdale from the side of a minister, Chillingworth from a more secular side. The one advantage Chillingworth of course has, is that he knows Dimmesdale harbors a secret. He does his best to hide this knowledge, but at the same time, does what he can to be a constant reminder to Dimmesdale of his sin and guilt.
Dimmesdale's physical health weakens as a result, and Chillingworth takes care of him. In this way, he is convincing himself that this "favor" he is doing is for the minister's own good.
Make no mistake. Chillingworth's actions are simply part of an evil and twisted revenge plot which victimizes Dimmesdale until the very end of his life.