Conscience probably interested Shakespeare because it gave him a chance to delve into the human mind. Conscience can work in two ways - in the decision makng process before an event and in producing feelings of guilt after the event in a wrong choice was made. We see both these aspects at work with a vengeance in Macbeth. In the speech featured on the Macbeth page of this site we see Macbeth wrestling with his conscience - in his ambition to become King he is open the considering the option of murder. His well informed conscience provides him the reasons why he shouldn't murder - Duncan, the current King is his cousin, guest and King, and a good king at that. But what really seems to bother him is the prospect of getting caught which suggests that his conscience is rather flexible! Lady Macbeth, recognises this - while she reckons he wouldn't do wrong ("too full o' the milk of human kindness"), he might be prepared to accept the benefit if someone else (like herself) did the wrong for him. Eventually they do the murder, and Macbeth murders plenty more, but conscience fills them with guilt after the event - Macbeth becomes increasingly unstable and paranoid and Lady Macbeth can't sleep peacefully anymore.
Hamlet doesn't seem to have any qualms of conscience about seeking revenge on Claudius, suggesting a non-Christian worldview, though he is reluctant to do it unless he is sure he has got the right culprit. At one stage he delays because he is afraid that the ghost might be a demon sent to lead him astray. If anything his conscience seems to be telling him that it is his duty to get revenge. Early in the play he is reluctant to solve his problems by committing suicide, because it's against God's law: "oh that the everlasting had not fixed his canon 'gainst self-slaughter".
In Merchant of Venice Portia appeals to Shylock's conscience, trying to get him to show mercy to Antonio, but without success. Shylock's conscience has been hardened by bitter experience.