The witches do have something to do with it. Macbeth has no plans to kill the king or wishes for the King's death until seeing the prophecies of the witches. Had the battle against Norway ended and Macbeth and Banquo gone home without seeing the witches, King Duncan probably would not have been killed. Upon learning that the second prophecy was true, then Macbeth began considering the third prophecy. In fact, even upon hearing the prophecy for the first time, Macbeth replies by saying, "to be king stands not within the prospect of belief."
Then, after learning of his new title, he says,
"Glamis, and Thane of Cawdor! The greatest is behind."
If the witches wouldn't have said anything to Macbeth, he probably wouldn't have even considered becoming king. On the other hand, they prophesy that it will happen, but they do not make it happen. Macbeth's choices might have single handedly caused the death of the King. This play brings up the debate of free will versus fate, but Shakespeare seems to support free will ruling over fate. They predict that he will be king, which is what leads him to actually think properly about killing Duncan, and of course what prompts Lady Macbeth to insist that he carries out the crime. But, it seems, his "black and deep desire" for the throne was there before the witches’ prophecies too. They might catalyse him into action. But he does it.
Lady Macbeth also has something to do with it. When Macbeth backs out of doing the deed, she persuades him to do it again and without her, he probably wouldn't have gone...