Explain how the character of Macbeth changed after the killing of King Duncan.

Prior to killing Duncan, Macbeth felt guilty and unsure.  He told his wife, Lady Macbeth, "We will proceed no further in this business" (1.7.34), and he even hallucinates a dagger, first clean and then bloodied, because he is so anxious and guilt-ridden at the thought of the murder he's about to commit.  However, once Macbeth has done the deed, and after he gets over his initial concern for his soul, he becomes a great deal more ruthless and feels a lot less guilty about the violent steps he takes. 
Once he's killed Duncan in order to acquire the throne, he soon begins to feel that he must get rid of Banquo and Banquo's son if he is to keep his new position and power.  The Weird Sisters had said that Banquo would not be king, but he would father kings, and Macbeth doesn't want to lose the throne to anyone of Banquo's line.  He promptly decides to kill his once-best friend; he convinces two murderers that it is Banquo's fault that they are poor and destitute, and thus persuades them to murder him (and Fleance, though Fleance escapes).  Further, Macbeth feels no guilt about this action, and he no longer considers the state of his soul. 
Descending into even greater ruthlessness and tyranny, Macbeth eventually murders the family and servants of Macduff, as a way of sending a message to his political enemy.  Macbeth has murdered Duncan and Banquo, and attempted to murder Fleance, and now he actually murders an innocent woman and her children as well as a great many innocent servants, simply because of their connection to Macduff.  It is a despicable move that showcases just how much worse a person Macbeth has become; he's abandoned every loyalty and he seems to have no conscience left.