Shakespeare first presents Macbeth's decision to kill Duncan through a series of asides. An aside is a dramatic convention where a character speaks to himself, another character, or the audience, without being heard by any other character on stage.
When he learns that he has been named Thane of Cawdor, just as the Weird Sisters predicted, he says, in an aside to himself,
If chance will have me king, why, chance may
Without my stir. (1.3. 157-159)
He hopes that since the first title fell into his lap that the second title will too. Wouldn't it be great if he didn't have to lift a finger to become king? He has not yet considered what he would do if this doesn't happen. Once Macbeth learns, however, that Duncan has named his son, Malcolm, the heir to his throne, Macbeth's thoughts quickly turn dark. In another aside to himself, he says,
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. (1.4.56-59)
Macbeth asks the stars to put out their lights so that no one will be able to see that he is thinking evil ("black") thoughts. He will not let his eye look to see what his hand is doing, and yet he is still going to let his hand do whatever it is that his eye would be afraid to watch. In other words, he's resolved to kill Duncan, even though he already feels sort of guilty about it (he doesn't want his eye to see the murder that his hand commits).
When he arrives home, conspiring with his wife leads to an actual plan to kill the king. When she asks him when Duncan plans to leave their home, Macbeth says, "Tomorrow, as he purposes" (1.5.70). In other words: tomorrow, or so he thinks [insert evil laughter here]. Though neither of them has actually spoken the words aloud, it is clear that they are on the same page; they don't even need to discuss their intentions because they know each other so well and can tell what the other is thinking.
Although Macbeth will go on to waffle a bit (resolving not to kill Duncan and then being shamed into it by Lady Macbeth), he eventually does go through with it. The part that always shocks me, though, is how quickly he transitions from loyal, honored subject to traitorous, regicidal murderer.