who's really driving the ambitions of killing the king?

The true "brains behind the operation" would have to be Lady Macbeth. I only say this because Macbeth does want to become king but he has no kind of backbone. So in turn she actually decides to convince her husband to kill the king. If she comes in and takes controll then it would be her running the country not her husband. Like Hilliary and Bill as they say. Hilliary Clinton ran the country not Bill. The same would be with Lady Macbeth and Macbeth.