Who is responsible for Macbeth's downfall and his death?

Macbeth is wholly responsible for his destiny. He should take the blame for whatever happened to him. The fact that he was driven by ambition was a choice he made. 

When we meet him, we hear about a soldier who is loyal and courageous. A man who was driven to defeat his country's enemies and utterly destroy them, whatever it took. He receives generous praise and is rewarded for his noble actions. 

We soon learn, however, that he harbours a pernicious desire to achieve the Scottish throne by illegitimate means. He admits that he is driven by 'vaulting ambition' which clearly indicates that he does not wish to ascend by having to follow the natural rules of succession. His desire is to become king as soon as possible, by hook or by crook. When he and Banquo meet the witches and they greet him by a new title as well as inform him that he will be 'king hereafter', he is sceptical until Ross and Angus arrive.

When Ross informs him that he has been awarded the title thane of Cawdor as the witches predicted he says, in an aside:

Glamis, and thane of Cawdor!
The greatest is behind.

This clearly indicates Macbeth's intent. He believes that he has now overcome one of his greatest hurdles. Being bestowed with such a great title brings him closer to the throne. This is emphasised in a later aside:

Two truths are told,
As happy prologues to the swelling act
Of the imperial theme ...

Macbeth's statement makes it obvious that he sees the witches' accurate predictions as an indication that destiny favours him. The idea of being king grows ever stronger. It is significant how he and Banquo differ in their perceptions of the witches. Whilst Banquo remains sceptical and expresses his cynicism, Macbeth is 'rapt', overwhelmed by what has happened.

Since he now believes that he is destined to be liege (thou shalt be king hereafter) Macbeth begins to plot his ascension. He informs Lady Macbeth of the good news and on his arrival back at his castle, he and his wife begin plotting King Duncan's murder. Macbeth expresses doubt about their plan to assassinate the king on a few occasions but is easily persuaded by his malevolent and ruthless wife to press on. If he had refused, the murder would not have happened.

Even though Macbeth is anxious and under great emotional and mental strain about the terrible deed he is about to commit, he does not stop. He eventualy cold-bloodedly murders king Duncan in his sleep and implicates his guards by planting evidence. He later murders them as well, completely removing any chance that he might be implicated.

Duncan's sons, Malcolm and Donalbain, flee in fear of their lives and suspicion for the murder falls on them. Without a closer relative to succeed to the throne, Macbeth is crowned at Scone. 

Once he gains the title, Macbeth goes on a murderous rampage. He kills his friend and confidante, Banquo by sending murderers after him, suspecting that Banquo is a danger. The assassins, however, fail to kill his son, Fleance, who escapes. Macbeth then consults the witches again who, by using paradoxical and equivocal statements and predictions, warn him about Macduffe and encourage him. Their assertions that he will not be harmed by one of woman born and that he shall not be vanquished until great Birnam Wood come to Dunsinane, inspires Macbeth. He feels invincible and sets out to destroy Macduff. He sends his assassins to Macduff's castle where they murder his entire family as well as his servants.   

We read about Macbeth's paranoia and his ruthless attempts to execute all those he sees as a threat. He becomes so 'steeped in blood' that he acknowledges that there is no turning back and, therefore, continues with his murderous rampage. Malcolm has, in the interim, obtained the assistance of the English king, Edward, to overthrow Macbeth and he is joined by Macduff who is later outraged when he learns about his family's brutal and senseless murder. He swears revenge.  

In the final act, Macbeth is at first, informed by a messenger that Birnam Wood was moving towards his castle. This is true since Malcolm's troops were disguising their numbers by each bearing a branch in front of him. Macbeth slowly realises that the witches had tricked him. When he is confronted by Macduff, he shockingly learns just how malevolent they had actually been, for Macduff tells him that he was 'from his mother's womb untimely ripped' and thus not, in the true sense of the word, 'of woman born'.

Macduff kills Macbeth in battle and beheads him. The tyrant has been vanquished. Macbeth had become a victim of his own ambition, gullibility and malice. He made the wrong choices and believed the wrong people. He alone was responsible for his downfall and eventual death.