How does Macbeth change throughout the play? What influences him to change?

Shakespeare needed to have Macbeth change into a tyrant for the purposes of his plot. Macbeth's tyranny and rule by terror makes many Scottish nobles and commoners flee to England. This unrest in Scotland is seen as a threat to the stability of his own realm by the English king, and this is what motivates him to go to the expense of raising a big army to overthrow the tyrant. If Macbeth had been a kind and efficient ruler, the English king would probably not have interfered in Scottish politics in spite of the pleas of Malcolm and Macduff. The fact that Malcolm was the heir apparent to the Scottish throne would not have been sufficient in itself to move the English king to help him.

King Edward knows nothing about what really happened when Duncan was murdered. The story Macbeth tells is that Malcolm and Donalbain bribed Duncan's two grooms to kill the old man in his sleep so that Malcolm could become king and Donalbain could become next in line of succession and also profit by getting all sorts of endowments from his older brother. Then, according to Macbeth's story, the two brothers fled because they were afraid of being found guilty of murdering their father. King Edward would have no idea whether or not the story was true. For all he knew, Malcolm might have had his father killed so he could succeed him. But in any case, Edward would not feel justified in invading Scotland with what was then a huge army if Macbeth had turned out to be a wise and worthy ruler.

Macbeth's descent into tyranny is hard to explain. No doubt he had no intention of being a wicked ruler when he murdered Duncan. He probably hoped to salve his conscience by at least trying to be as good a king as Duncan had been. He says in a soliloquy in Act 5, Scene 3:

I have lived long enough. My way of life
Is fall'n into the sear, the yellow leaf,
And that which should accompany old age,
As honor, love, obedience, troops of friends,
I must not look to have; but, in their stead,
Curses, not loud but deep, mouth-honor, breath,
Which the poor heart would fain deny and dare not.

That must have been what he had hoped for if he became king--honor, love, obedience, troops of friends. He certainly couldn't have been hoping for what he actually got. The only plausible explanation for why he became such a hateful tyrant is that everybody knew he had committed the worst sort of treason and were recalcitrant because he was not the legitimate ruler. This then compelled him to rule by force and terror. Otherwise he could not get obedience. But the force and terror only made everyone hate him all the more.