In "The Scarlet Letter" how does Chillingworth know that Dimmsdale is keeping a secret?
He doesn't know for sure, at first. But as he spends more and more time with Dimmesdale, he notices symptoms that lead him to conclude that Dimmesdale is tortured with something. He is sickly; he flushed often; he is quiet and pained; he has strange injuries. All of these things lead Chillingworth to conclude that Dimmesdale is not emotionally well. He is burdened with something, and since Chillingworth is on the hunt for the man who has not confessed to adultery with Hester, he draws conclusions. He hopes. He suspects that Dimesdale might be the one. So, he spends more and more time with him, and watches and observes him closely. Pretty soon, he is the reverend's personal physician, and even moves in with Dimmesdale as his health falter. In their conversations, Chillingworth desperately and subtly tries to get Dimmesdale to slip up, to reveal something, or perhaps to even confess his crime, to no avail.
It is while Dimmesdale is sleeping one night that Chillingworth's supsicions are confirmed; he looks at the sleeping Dimmesdale, and sees something on his chest that verifies he is indeed the man Hester had been with. There, on Dimmesdale's chest, is an engraved letter A, cut into his skin and scarred over. Dimmesdale, wracked with guilt, had been trying to purge his guilt by tatooing a letter A in his own chest. He was "confessing" to God and himself and trying to make up for his sin in his own way. This confirms Chillingworth's suspicions that Dimmesdale's awful secret is that he is Pearl's father and Hester's lover. I hope that those thoughts helped; good luck!