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,...

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:

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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.