What is leech? What double meaning does the world "leech" have?
A leech is a blood-sucking parasite, and it is also used to describe a person who sponges off or uses other people. Therefore, it is a perfect word to describe Chillingworth because he is figuratively sucking the life out of his "host" (so to speak), Arthur Dimmesdale. Dimmesdale is getting weaker and weaker, as though his life is ebbing away. Chillingworth believes, correctly, that Dimmesdale is Hester's co-sinner and the father of little Pearl, and he has vowed to ruin Dimmesdale and make him pay for the harm that the minister has done to him (by sleeping with his wife). Chillingworth is sort of living off Dimmesdale, his hatred and need for revenge keeping him alive, as, when Dimmesdale dies, Chillingworth dies shortly thereafter (just as a parasite would not be able to survive without its host).
Doctors in that time period were also sometimes called leeches because they would use leeches to bleed their patients. They would attached the leeches to a person to try and extract 'bad blood' that they thought was making the patient sick.