He decides to kill Duncan in the first act, and his quote that supports his fears is in Act I, sc 4. He is not happy that Malcolm is next in line, so he must move quickly if he is to kill Duncan.
The Prince of Cumberland! that is a step
On which I must fall down, or else o'erleap,
For in my way it lies. Stars, hide your fires;
Let not light see my black and deep desires:
The eye wink at the hand; yet let that be,
Which the eye fears, when it is done, to see.
Then in Act III, sc 1, he asks Banquo where he's riding, when he'll be back, and if his son is going. He is setting up the murder of Banquo at this point. This is his quote from that scene:
To really understand Macbeth's motivation for committing these murders, take a look at Act I, Scene III, in which the witches make three prophecies. According to the second of these, Macbeth will become king:
All hail, Macbeth, that shalt be king hereafter.
In addition, according to the third, Banquo's sons will rule as kings:
Thou shalt get kings, though thou be none.
These two quotes are important because they ignite Macbeth's deepest desire for power. Macbeth wants these prophecies to come true and, as shown later in the scene, he begins to dwell on this "imperial theme."
Later, in Act I, Scene V, Macbeth writes a letter to his wife in which he reveals the prophecies. Notice how he talks about their exciting future ("greatness") in which he is certain to become king (if Duncan is removed):
This have I thought good to deliver thee, my dearest partner of greatness, that thou might’st not lose the dues of rejoicing, by being ignorant of what greatness is promised thee.
Then, in Act I, Scene VII, Macbeth refers again to his desire to be king, specifically his ambition:
I have no spur
To prick the sides of my intent, but only
Vaulting ambition, which o'erleaps itself
And falls on th' other.
So, it is clear that his motivations for killing Duncan are all based on his ambition to rule Scotland and to provide a great future for himself and his wife.
Similarly, on the issue of killing Banquo, take a look at this quote from Act III, Scene I, in which Macbeth expresses his fear that the above prophecy will come true and that Banquo's sons will rule Scotland. Macbeth, therefore, must act to protect his regal position:
Our fears in Banquo
Stick deep, and in his royalty of nature
Reigns that which would be feared.
Finally, in the next few lines, Macbeth makes it clear that he must eliminate Banquo and his sons if he is to remain king:
To make them kings, the seed of Banquo kings!Rather than so, come fate into the list,And champion me to th' utterance.