One quote that captures the essence of Act III are the words of Macbeth:
"It will have blood, they say: blood will have blood" (3.4.129)
In Act III, Macbeth's hands are figuratively covered in blood and he becomes more violent, and more horrified as he prophesies of the witches constantly looms over Macbeth. After killing Duncan, Macbeth worries that the next prediction of the witches, the one that Banquo will have sons who will be kings, will come true:
To be thus [King]is nothing, but to be safely thus--
Our fears in Banquo stick deep.
And in his royalty of nature reigns that
Which would be feared. (3.1.50-53)
Therefore, in order to eliminate Banquo, he calls two of his men, reminds them of wrongs that Banquo has done against them and convinces them to murder him. The men leave and later attack Banquo and his son Fleance, killing Banquo, but Fleance escapes. After learning of the murderers' activities, Macbeth worries that he has merely "scotched the snake, not killed it"; thus, he fears yet the prediction of the witches. Lady Macbeth tells him "What's done is done," (3.2.14),but Macbeth moves toward suspicion and paranoia about Banquo's son Fleance.
Each murder that Macbeth commits does not finish anything; they are not "done." Instead, Macbeth worries who next will stand in his way, saying "blood will have blood," meaning the dead will find their revenge (this is why he sees Banquo's ghost in Scene 4). So, Macbeth feels that he has gone so far that he can only continue in his bloody deeds and violent action in order to remain king.
I am in bloodStepped in so far that, should I wade no more,Returning were as tedious as go o'er. (3.4.142-144)