In act 1, scene 4, King Duncan names Malcolm to be his heir, which influences Macbeth to begin thinking about committing regicide. Macbeth acknowledges that in order to fulfill the prophecy and become king of Scotland, he must assassinate Duncan. In an aside, Macbeth says,

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.

At the beginning of act 1, scene 7, Macbeth lists the numerous reasons why he should not assassinate King Duncan and admits that the only thing motivating him to commit regicide is his ambition. Macbeth says,

I have no spur
To prick the sides of my intent, but only
Vaulting ambition, which o'erleaps itself
And falls on th' other.

Following King Duncan's assassination, Macbeth desires to cement his legacy as king and begins to plan Banquo's murder. Macbeth does not want Banquo's prophecy to come to fruition and is willing to kill his former friend in order to prevent Banquo's descendants from attaining the throne. In act 3, scene 1, Macbeth says,

To be thus 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. 'Tis much he dares,
And to that dauntless temper of his mind
He hath a wisdom that doth guide his valor
To act in safety. There is none but he
Whose being I do fear, and under him
My genius is rebuked, as it is said
Mark Antony’s was by Caesar.

Macbeth is expressing his fear of Banquo during this soliloquy and feels that he has doomed his soul for nothing if Banquo's descendants eventually attain the throne. Therefore, Macbeth is motivated to kill Banquo and Fleance to protect and cement his legacy as king of Scotland.

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...