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!
I would encourage you to remember that Hawthorne went to great efforts to find documents supporting the historical truth to the situations that he put together in this piece. To create the character of Chillingsworth, he must have found some very incriminating ideas. See what you can find out and it might spark your own points.
Another idea would be to look at Chillingsworth (since your question says role) as the different roles he may have played in the story. At times he was indeed the physician. But he also played the role of jealous husband, a self-proclaimed judge, and a citizen of the town. There might even be another role or two you can think up for him. Once you determined three roles, it would be easy to find quotes that determine his flaws and strengths at each role.
Three good points to make about Chillingworth are basically how his life changed before, during, and after the incident with Hester.
For the before part, you could research on what was his job, and speculate about his lifestyle prior to sending Hester away to America.
For the during part, you can explore and extrapolate how he began to change, went undercover under a different name to Boston, how he infiltrated the life of Dimmesdale, and how he began to haunt him to his death.
For the after part, point out how the deat of Dimmesdale still did not quench the thirst of revenge that he had, and that after one year he died accomplishing absolutely nothing.