To get him to admit that he slept with Hester.
Roger Chillingworth wants revenge on Reverend Dimmesdale for having an affair with Hester Prynne. He seeks to torment Dimmesdale as he believes Dimmesdale is partly responsible for the shame and suffering that Hester endured. Chillingworth aims to expose Dimmesdale's guilt and hypocrisy.
Hester Prynne reveals to Reverend Dimmesdale that Roger Chillingworth is her husband. She discloses their secret marriage and the fact that Chillingworth is seeking revenge on Dimmesdale for his adultery with Hester.
If I'm not mistaken, Hester did not meet Chillingworth in the forest. She met him in the prison and on a beach (I believe). During the first interaction, he asked her to pretend she didn't know him. During the second, she asked him to stop torturing Dimmesdale.
Chillingworth was gone for about two years before Hester cheated on him with Dimmesdale.
Roger Chillingworth approaches Dimmesdale just as he invites Pearl and Hester to join him near the scaffold.
Chillingworth is pretty much a leech attached to Dimmesdale. Chillingworth is living and thriving off the pain/guilt he is inflicting on Dimmesdale to the point where he can no longer live after Dimmesdale's death.
Dimmesdale is shocked and horrified when Hester reveals Chillingworth's true identity as her husband. He feels betrayed and manipulated by Chillingworth for disguising his true relationship to Hester. Dimmesdale also feels guilt and remorse for unknowingly being involved in the scheme that has harmed Hester.
Chillingworth visits Hester in prison to confront her about her relationship with Dimmesdale and to assert his control over her. He wants to extract information from her and further manipulate the situation to seek revenge on Dimmesdale.
Dimmesdale realizes Chillingworth is not the friend he thought he was. He decides to dismiss Chillingworth later on and not take in his remedies. Hester basically told him that Chillingworth is her husband and he's out to discover his secret.
Chillingworth notes that Dimmesdale's defense of Hester during her public shaming is a sign of his inner torment and guilt. He sees it as a sign of Dimmesdale's own involvement in the sin that they share with Hester.
Chillingworth blames himself and the man whom Hester cheated with. He claims he should have forseen this, and the other man will eventually be revealed.
Chillingworth wanted to know the name of the man with whom she committed adultery, Dimmesdale.