Why is the relationship between Macbeth and his wife so important?
The relationship between Macbeth and Lady Macbeth is important for a couple of reasons. First, it seems unlikely that Macbeth would have actually gone through with the murder if it weren't for his wife. In Act 1, Scene 7, Macbeth contemplates all of the reasons he has not to kill the king: Duncan is his leader, his friend, his kinsman, his guest, and so on. He has only one reason to go through with it: his "Vaulting ambition" (1.7.27). Therefore, he tells his wife, "We will proceed no further in this business" (1.7.34). Lady Macbeth, however, is not so easily dissuaded from her purpose; she proceeds to verbally abuse him, telling him that he'll have to "live a coward in [his] own esteem" for the rest of his life, calling him "green and pale" (1.7.47, 41). She berates and insults him, saying that he isn't really a man if he doesn't stick to their plan, and, eventually, she convinces him to recommit to it. Thus, we might say that Macbeth never would have committed the murder if not for the Lady Macbeth.
Moreover, their relationship acts as a sort of yardstick by which we can measure their change in the play. By Act 3, Scene 2, Lady Macbeth no longer seems to be in the loop. Macbeth has planned the murder of Banquo and Fleance without her counsel, even without her knowledge. In fact, when he tells her how to treat Banquo at their dinner party later, he's lying to her because he knows that Banquo will not be there (because he'll be dead). Lady Macbeth is no longer happy, and neither is her husband, but they don't seem to commiserate in the way that they used to do. She is upset that they've worked so hard only to live in "doubtful joy," and he feels that "To be thus is nothing, / But to be safely thus" (3.2.9, 3.1.52-53). Neither is content, and perhaps if they worked together, they could figure out a solution, but they do not seem to speak honestly to one another anymore. It's a really significant change from Act 1.
By Act 5, Lady Macbeth is so distraught that she has begun to walk in her sleep, having terrible nightmares and flashbacks to the night of Duncan's murder. She even seems horrified, now, by the monster she has created out of her husband, lamenting, "The Thane of Fife had a wife. Where is she now?" (5.1.44-45). Macbeth ordered the brutal murders of Lady Macduff and her children, for no other reason than to wound Macduff. There was nothing to be gained by their deaths as there was with the murders of Duncan and Banquo. Macbeth has simply become so brutal that he acts without her knowledge and in ways, now, that not even she can condone.
Her eventual suicide is yet another clue as to how far apart they've grown. We see no conversation between them at all after Act 3, Scene 4 (the banquet scene), and the sheer lack of interaction between the couple further indicates how much they've changed. Lady Macbeth used to be his "partner of greatness," and she seems almost nothing to him now (1.5.11). He instructs the doctor to "Cure her" of whatever ails her but never once goes to her himself (5.3.49). In many ways, they have switched roles: she was once ruthless and hard, and now she has become guilt-ridden and soft; he was once guilt-ridden and has now become ruthless. Thus, their relationship helps us to see just how much they've changed throughout the course of the play.