Macbeth is aware that Banquo is getting suspicious, and he wants to get him out of the way.
As we can tell from Banquo’s speech at the beginning of the scene, he is getting suspicious. This is what he means when he says “Thou hast it now … and, I fear,/Thou play'dst most foully for't” (Act 3, Scene 1). In other words, Macbeth has the position of king, but he did not get there in the right way. He got there by murdering Duncan. Macbeth explains that to be king is nothing unless he is “safely thus.”
fears in Banquo
Stick deep; and in his royalty of nature
Reigns that which would be fear'd: 'tis much he dares (Act 3, Scene 1)
So Macbeth invites some murderers in. He does not want to do the dirty work himself this time. It is better to be totally above suspicion. He will hire some thugs to waylay Banquo and Fleance, his son, when they are out riding while Macbeth and his guests are in the castle.
Macbeth explains that Fleance has to die with his father because he will be with him. Of course, if Banquo is killed it is expedient to kill Fleance because he might try to avenge his death. More importantly, the witches said that Banquo's sons would be kings. We can't have that! Banquo and his heir are a threat to Macbeth and his line.
his son, that keeps him company,
Whose absence is no less material to me
Than is his father's, must embrace the fate
Of that dark hour. (Act 3, Scene 1)
Fleance and Banquo are both murdered, but Banquo makes a surprise appearance as a ghost at the banquet. Macbeth’s guilt, or madness, makes him crack when he tries to sit down and seems to find Banquo there! He replies that the “table's full” (Act 3, Scene 4), and no one else can see the ghost of Banquo. Lady Macbeth makes apologies and is greatly embarrassed.
Banquo's and Fleance’s deaths transforms Macbeth from a murderer to a bloodthirsty tyrant. He is desperately trying to hang on to his power in any way he can, now that he has it. He feared Banquo, his best friend, or at least his close friend, would betray him. Banquo was suspicious, and was there in the beginning when the witches gave their prophecy. He knew that they said that his sons would be king. He had to go.
I think you ought to include most of Macbeth's soliloquy in Act 3, Scene 1 in your essay. This is the pertinent part:
He chid the sisters,
When first they put the name of King upon me,
And bade them speak to him; then prophet-like
They hail'd him father to a line of kings:
Upon my head they placed a fruitless crown
And put a barren sceptre in my gripe,
Thence to be wrench'd with an unlineal hand,
No son of mine succeeding. If't be so,
For Banquo's issue have I filed my mind,
For them the gracious Duncan have I murdered,
Put rancors in the vessel of my peace
Only for them, and mine eternal jewel
Given to the common enemy of man,
To make them kings, the seed of Banquo kings!
Rather than so, come, Fate, into the list,
And champion me to the utterance!
There is some beautiful language in this soliloquy. Two of the most striking things, in my opinion, are the way he acknowledges that he has sold his soul to the devil without using those exact words. He doesn't even want to think about what awaits him in the life to come. The words "Given to the common enemy of man" seem to need to be spoken hastily, as if he feels he has to say something about the matter but doesn't want to dwell on it. He has sold his soul to the devil, the worst thing a man could possibly do, solely in order to make it possible for Banquo's descendants to become kings of Scotland. He must suspect that Banquo finds this secretly amusing. All Banquo has to do is watch and wait, and everything will be taken care of for him by Fate.
The other striking notion in this soliloquy is that Macbeth actually decides to challenge Fate. He assumes Fate has decreed that Banquo's descendants will be future kings, but he won't accept this. He will kill Banquo and Fleance and make it impossible for Banquo to have any descendants. Then how could things work out the way the Three Witches foretold? They are only speaking for Fate. They have no power to dictate the future themselves. Macbeth's decision to battle with Fate itself is perhaps the only admirable and truly heroic thing about him. And this may be why the play is a tragedy. Macbeth is obviously the protagonist, and Shakespeare must have seen Fate as the antagonist. Macbeth personifies Fate as a knight in armor--perhaps a black knight with his face hidden by his helmet--against whom he must fight in a tournament with lances and swords. This is the sort of thing Macbeth understands and enjoys. He is a warrior.
It occurs to me that, depending on how much space you have, you might quote some of the things said by the Three Witches, such as the prophecies they make to Macbeth and Banquo when they first meet.