This moment of resolve happens at the end of act 2, scene 1. As Macbeth focuses his efforts on ridding himself of Duncan, he becomes agitated, even seeing things that aren't there:
Is this a dagger which I see before me,
The handle toward my hand? (2.1.41–42)
As he tries to grab it, his hand passes right through. Some literary critics point to this as when Macbeth begins to lose his sanity and spiral out of control in his actions. Others argue that he is very much in control, continuing to make decisions with a declining moral compass. Either way, he is clearly not calm as he approaches his intended victim, speaking to the floating apparition of a dagger as he proceeds.
Macbeth finally seems to snap back into reality with a moment of clarity:
There’s no such thing.
It is the bloody business which informs
Thus to mine eyes. (2.1.55–57)
As the scene concludes, he finds his strength and clarity of purpose, asking that his steps be light enough to not be heard so that he can complete the murder he has set out to do. He wishes to make haste with his actions so that he doesn't lose the passion and courage that is necessary to kill Duncan.
His final speech in this scene begins with hesitation and an overwhelming sense of guilt that seems to manifest itself in the vision of a bloody dagger. However, Macbeth's speech shifts near the end to that of a cold and calculated killer who is intent on murdering this good and faithful king.