What is the relationship between Macbeth and Duncan?
First and foremost, Macbeth is Duncan's loyal subject, or, at least, he is at the beginning of the play. When first we see Macbeth and Duncan interact in Act 1, scene 4, Macbeth tells his king, "The service and the loyalty I owe / In doing it pays itself" (1.4.25-26). In other words, it is Macbeth's duty to serve Duncan, and doing so is its own reward. As Duncan's subject, Macbeth is obligated as a child is to a parent or a servant to a master.
Next, Duncan and Macbeth are family. As soon as Macbeth enters the room in this scene, Duncan greets him, calling him, "worthiest cousin" (1.4.17). In addition, Duncan feels indebted to Macbeth for his cousin's tremendous defense of Scotland against both a rebel as well as a foreign invader. He says, "From hence to Inverness / And bind us further to you" (1.4.48-49). Duncan tells Macbeth that he and his retinue will go to Macbeth's castle, and this visit will cement their relationship even further.
On the night of the murder, Macbeth references both of these relationships, saying, "I am his kinsman and his subject, / Strong both against the deed" (1.7.13-24). In other words, both of these relationships should prevent Macbeth from even considering killing Duncan. Further, now that Duncan is at Macbeth's home, relying on Macbeth's honesty and hospitality, Macbeth is "his host, / Who should against his murderer shut the door, / Not bear the knife myself" (1.7.14-15).
Duncan and Macbeth are cousins. In Act 1, sc. 7, in Macbeth's soliloquy, he says, "...He's here in double trust:/ First, as I am his kinsman...". It is this family relationship that, in part, allows for him to become king. During this soliloquy in Act 1, sc. 7, Macbeth says that he has no reason to kill Duncan except for his "driving ambition". He says that he is kin to Duncan and he is Duncan's subject which are two excellent reasons to not kill him. Of course, it doesn't take his wife long to convince him otherwise. Not only would the new king of England, King James I have been flattered by the idea that it was wrong and upset the natural order to kill a king, but he would also liked the idea being presented that it was unnatural to kill a relative. That it is unnatural to kill a relative is stated in Act 2, sc. 4 by Ross when he says, in reference to Malcolm and Donalbain having fled Scotland presumably because of their guilt, "Gainst nature still!"