Macbeth persuades the murderers to kill Banquo by telling them that Banquo is their enemy.
When we first see Macbeth with the murderers, he reminds them that he has already talked to them and explained everything to them. Banquo is their enemy, not Macbeth.
That it was he, in the times past, which held you
So under fortune, which you thought had been
Our innocent self? This I made good to you
In our last conference (Act 3, Scene 1)
Macbeth doesn’t explain exactly what it was that Banquo was supposed to have done to these murderers that made them so “unfortunate.” Apparently he has been up to no good. Shakespeare is a little vague one this. It doesn’t matter. I am pretty sure the murderers just want to get paid, and Macbeth is the king after all. They are going to do what they are told. Macbeth says Banquo is a bad guy, so Banquo is a bad guy. Murderers tend to be not too particular in these matters.
Macbeth explains that he knows what kind of people they are, and he understands them. He says that he is sick while Banquo is alive, and will be healthy when he is dead. He then goes on to make it very clear to them.
Both of you
Know Banquo was your enemy.
True, my lord.
So is he mine, and in such bloody distance
That every minute of his being thrusts
Against my near'st of life…(Act 3, Scene 1)
Since they are murderers, he asks for their help, saying, “I to your assistance do make love.” Of course, they say they would be happy to oblige. What are friends in low places for? Macbeth gives them the details and makes sure they know that they need to kill Fleance too, because Banquo’s son’s absence is “is no less material” to him than his father’s.
There are so many details to a murder! Macbeth should know that by now. His murderers kill Banquo successfully, but Fleance gets away, leading them to say, “We have lost best half of our affair” (Act 3, Scene 3). Maybe Macbeth should not have planned everything himself, and left it to the professionals. You kill one guy successfully and you think you’re an expert!
Macbeth killed Banquo because of the prophecy that his sons would be king, and because he thought Banquo was suspicious that Macbeth got his crown through less than proper ways. Macbeth was right. Banquo was suspicious. Banquo's death will just be one in a long line for Macbeth, who finds that killing once is not enough to keep his throne.