How is Macbeth responsible for the tragic outcome of the play and ultimately his own downfall?
Macbeth had it prophecied that he would become Thane of Cawdor, and ultimately King of Scotland, by the witches in Act I. When Macbeth and Banquo then rendezvous with King Duncan, the King informs Macbeth that he has been made Thane of Cawdor, replacing the traitor who had shown his hand in a recent battle. Then King Duncan names his son as Prince of Cumberland.
Here Macbeth has a choice. If he is to become King of Scotland, he must somehow become Prince of Cumberland, or usurp the Prince. If he trusts the witches' prophecy, then he need do nothing. Fate should ultimately lead to both King Duncan and his son being dethroned, and the only Thane with two Thanedoms would then be the logical choice to become King. Or he can hasten the inevitable, which he does, choosing to murder King Duncan, and frame the Prince thus removing both the King and his heir.
Once Macbeth becomes King, he remembers that the Witches also prophecied that Banquo would father a line of Kings. This suggests that Macbeth's descendants will not remain on the throne. This prophecy may be fulfilled within Macbeth's reign, or may be fulfilled many generations from now. Again, Macbeth has the choice of trusting whether it will happen in the irrelevant future, or pose an immediate threat to his rule. Macbeth chooses to assume that Banquo's lineage poses a direct threat to his rule, and thus dispatches assassins to kill Banquo and his son. This begins to seal his reputation as a tyrant.
Later Macbeth goes to see the Witches again, to see how long his rule will last. The Witches tell him to beware of Macduff, the Thane of Fife, that his reign will last until the forest comes to the gates of the Castle, and that he need not fear "a man of woman-born." These prophecies have multiple meanings too. Macduff poses a possible threat to his rule, but it is up to Macbeth how to deal with this threat. The forest coming to the Castle would indicate a long reign, because a forest moves like a glacier advances (i.e. more trees grow closer to the castle, just as more ice accumulates at the top than melts at the bottom), but could also have a more figurative meaning. Not fearing a man of woman-born makes no sense, because all men are born of women, aren't they?
Macbeth makes all the wrong calls on this. Rather than leaving Macduff alone (which would fulfill being wary of the Thane of Fife), Macbeth orders Macduff and his family murdered, sealing his reputation as a tyrant, and earning Macduff's wrath. Macduff then joins the sons of Banquo and Duncan in a conspiracy to end Macbeth's Tyranny and restore the throne to the proper lineage. The forest prophecy is fulfilled when the conspiracy's invasion force uses Forest boughs as camouflage to approach the Castle, and the prophecy of no man of woman-born, is fulfilled by the fact that Macduff was born by C-section.
If Macbeth had not tried to hasten the process, and trusted in fate to fulfill the Witches' prophecies, just as it had fulfilled the prophecy of his becoming Thane of Cawdor, then he would have become King legitimately, might have reigned for a long time, and been succeeded by one of Banquo's descendants, but he chose to tempt fate, and thus sealed his doom as a short-lived tyrant.