Since Chillingworth's overwhelming motivation is revenge, you might consider how this desire for retribution impacts the three major characters: Hester, Dimmesdale, and Chillingworth himself.
In Hester's case, ask yourself how Chillingworth's relentless probing of Dimmesdale and the imminent threat of exposure leads her to act in ways that she might otherwise not have. It might be worth exploring how Chillingworth's machinations actually force Hester to face her past and act decisively. Does Chillingworth actually wind up doing Hester a favor in the end?
Dimmesdale's inner turmoil is stoked by Chillingworth's merciless cross-examination. The unhappy fate of the guilt-ridden reverend is no doubt hastened by Chillingworth, but you might want to ask whether the "physician" unwittingly cures his "patient" by compelling him to confess his transgressions and, ironically, win even more honor in the process. Dimmesdale's death is a sobering reminder of the inevitable consequences of the ruthless lust for vengeance.
Chillingworth pays a heavy price for his unwillingness to forgive and for the punishment he exacts from his cuckolder. A question I always raise with my students is the meaning of the bequest he makes to Pearl before he dies. Is it in any way a measure of redemption for the embittered old man? Or is he beyond redemption?
All three major characters are victimized by Chillingworth's obsession for payback. Just how this happens in each case provides you with plenty of room for insight and original analysis. Good luck!