What is Danforth's role in the proceedings? How would you characterize Danforth's personality?

Danforth, as we are told in some background information in act three, is the Deputy Governor of the colony. He is, evidently, the chief magistrate in the trials as a result of his position, and people present themselves and their evidence directly to him.
As for Danforth's personality, Miller says that "he is a grave man in his sixties, of some humor and sophistication that does not, however, interfere with an exact loyalty to his position and his cause." Danforth proves that he cares more about his authority and position than he does about finding the truth and protecting the innocent. In the court, he says things like, "We burn a hot fire here; it melts down all concealment." But, in the end, in act four, when he is presented with information that indicates that the convictions the court has made are actually in error—that those sentenced to die are actually innocent—he chooses to move forward with the executions rather than delay and risk undermining his own authority and power. To those who advise him to delay the hangings, he says,

Postponement now speaks a floundering on my part; reprieve or pardon must cast doubt upon the guilt of them that died till now. While I speak God's law, I will not crack its voice with whimpering. If retaliation is your fear, know this—I should hang ten thousand that dared to rise against the law, and an ocean of salt tears could not melt the resolution of the statutes.

Rather than risk undermining his authority, he sends people who he knows are very likely innocent to their deaths. Thus, he is more interested in protecting himself and his position than he is in finding the truth or protecting innocent people.