What conflict rages in Macbeth after he hears the witches prophecy? What resolution to this conflict does Macbeth express in his aside, in Act 1, Scene 4?
The conflict is that Macbeth is eager to become king, as the witches prophesy, but he sees no ready way to this achievement - except murder: the murder not only of the present king Duncan but his son and designated heir to the throne, Malcolm, named as Prince of Cumberland.
Macbeth abhors the idea of murder to achieve his ends yet cannot put it aside. This is the terrible conflict that takes place within him. He admits in an aside that he has been shaken to the core at the thought; his hair stands on end and his heart beats hard as he contemplates this unnatural deed (I.iii.134-138). Yet, because his ambition is so strong, he does contemplate murder. And he actually appears to reach the resolution that he will indeed commit murder to achieve his ambition, as evidenced in his aside in Act 1, sc. 4:
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. (50-53)
This means that although he is terrified to witness his own deed of murder, he will nevertheless perform it, while averting his gaze from it ('the eye wink at the hand'). He will be ashamed to look upon it but he still wants it to go ahead ('yet let that be/ which the eye fears').
It is interesting to note that Macbeth appears to reach this resolution of his own accord, before his wife appears on the scene. Lady Macbeth certainly urges him onto murder, but at this stage he seems resolved to perform it even without her encouragement.
Clearly Macbeth is excited to hear the prophecy about becoming king, but he has no understanding of how it could possibly come true. After the second prophecy of becoming Thane of Cawdor comes to being, he is even more enthralled with the possibility of being king. He admits to having a "horrid image" in his mind which scares him.
Thus, the conflict arises of how he will become king. He knows that he is, at best, third in line to the throne behind Duncan's sons. Therefore, he will have to commit a murder to become the king. This is the aforementioned "horrid image."
The reader knows that Macbeth is resolving his mind to committing a crime in his aside. In this scene, Duncan has just formally named his son Prince of Cumberland, indicating officially what most already knew - that Malcolm will be king next. After hearing his, Macbeth rages to himself:
The Prince of Cumberland! That is a step On which I must fall down, or else o'erleap, For in my way it lies.
This statement indicates his intention to act upon the prophesy, which he calls his "black and deep desires." He seems ready to kill.