How is Macbeth a tragic hero? To be specific how do his actions sequence to a catastrophe?
A tragic hero is a man (usually) of heroic qualities, generally a good person, who is brought low by his own error in judgment. We, the audience, are meant to pity him (and presumably learn from him).
Macbeth is a tragic hero because, as the Thane of Glamis, he is a noble man, a great warrior, proven in battle and admired and appreciated by his king (Duncan). In the opening scene, before we've even met him, the sergeant fresh from the battle tells Duncan that the rebel Macdonwald was faced down by "brave Macbeth" who "disdain[ed] fortune"--not caring whether he lived or died in the fray--"like valour's minion" hacked his way through the soldiers until he found Macdonwald and wordlessly unseam'd him from the nave to the chaps." Malcolm praises him for being valiant, and the sergeant adds that after Macdonwald's men fled, the Norwegian lord with whom he was allied, having fresh men and supplies, renewed the attack, but Macbeth was not dismayed; instead, he and Banquo doubled their efforts and waded into the battle. We see him being absolutely fearless in the protection of his king and his king's lands. He is noble and loyal.
When Macbeth learns from the witches that he is to be Thane of Cawdor (defeated in the battle he himself is fresh from) and "king hereafter," he does not believe it until Ross addresses him as Thane of Cawdor. At this point, Macbeth is excited and frightened, because he doesn't understand what it means. He admits that it seems like it's a good thing to have the first prophesy come true, then says:
If good, why do I yield to that suggestion
Whose horrid image doth unfix my hair
And make my seated heart knock at my ribs,
Against the use of nature? (I.3.)
He's already considering, deep down, what he might do next to become "king hereafter," but he knows it's unnatural and wrong.
When Duncan greets him and says he cannot thank him enough, Macbeth responds:
The service and the loyalty I owe,
In doing it, pays itself. Your highness' part
Is to receive our duties; and our duties
Are to your throne and state children and servants,
Which do but what they should, by doing every thing
Safe toward your love and honour. (I.4.24-9)
In so many words, he graciously tells the king that he and Banquo simply did their duty, and we have no reason to question his sincerity.
His first mistake is to tell his wife, via a letter, the Weird Sisters' prophesy. Lady Macbeth admits:
Glamis thou art, and Cawdor; and shalt be
What thou art promised: yet do I fear thy nature;
It is too full o' the milk of human kindness
To catch the nearest way: thou wouldst be great;
Art not without ambition, but without
The illness should attend it: what thou wouldst highly,
That wouldst thou holily; wouldst not play false,
And yet wouldst wrongly win: thou'ldst have, great Glamis,
That which cries 'Thus thou must do, if thou have it;
And that which rather thou dost fear to do
Than wishest should be undone.' (I.5.15-25)
In other words, He is Thane of Glamis and now Thane of Cawdor, and (if she has her way), he will be king (what he is promised). But, she admits up front that he is "too full of the milk of human kindness" to "catch the nearest way" (i.e., do it the easy way). She knows he is not the kind of man to cheat his way to the throne; the only way he would take is the holy path. Thus, she confirms that he is a good man, worthy of the honors heaped upon him.
Of course, Lady Macbeth has no such compunctions. She already knows she'll have to talk him into "catching the nearest way" by "chastising with the valour of [her] tongue."
In the beginning of I.7, Macbeth--having had the idea of murder planted in his head by his ambitious wife--muses about the consequences of it. He decides that he will not only have to pay for it by going to hell, but that such an action would be known in this life, as well, and he'd have to pay. He realizes that their plans are a "poison'd chalice" which they bring "To [their] own lips," then tells his lady that he will hear no more of the plan. She responds that he must not love her then and accuses him of being a coward. He responds with "Prithee, peace: / I dare do all that may become a man; / Who dares do more is none."
His weakness, then--or "tragic flaw," if you will--is the perception of others regarding his manhood. She sees this and accuses him not not being a man if he goes back on his vow to kill Duncan (a vow he did not make, incidentally).
He fears his own mental illness on the way to murder Duncan, but driven by his wife's insistence, goes and murders the king, re-emerging with the bloody dagger still in his hand. Lady Macbeth meets him and finds him troubled, complaining of hearing voices, and learns he has not "finished the deed"--that is, murdered the grooms, as well, and "gilded their faces" with Duncan's blood, to put the blame on them. Macbeth is too shook up to go back into the chamber, demonstrating that he is still the same man he was, regretting his deeds and now afraid for his soul, so Lady Macbeth does it for him.
He's begun his downward slide, so when Banquo muses to himself in III.1, "Thou hast it now: king, Cawdor, Glamis, all, / As the weird women promised, and, I fear, / Thou play'dst most foully for't," then tells Macbeth he must leave immediately, Macbeth realizes Banquo is onto him and must be silenced. He also is angry, now that the witches' prophesy is coming true, that Banquo is promised a line of kings while Macbeth himself will wear "a fruitless crown," and thus sends out two murderers to finish off Banquo. He admits to his wife that his mind is "full of scorpions"; he fears now the truth that everyone might see.
He is subsequently haunted by the ghost of his now-murdered friend, and plagued with fear of being found out. In III.4, he has come to fear what Macduff may think, as well, and makes plans to return to the weird sisters for more guidance:
And betimes I will, to the weird sisters:
More shall they speak; for now I am bent to know,
By the worst means, the worst. For mine own good,
All causes shall give way: I am in blood
Stepp'd in so far that, should I wade no more,
Returning were as tedious as go o'er.
The last lines of this passage is his acceptance that his deeds have led him to the point that he cannot back out of the situation now, so he may as well go on.
When he goes to the weird sisters, they summon three apparitions which tell him, respectively, that he is to fear Macduff, that no man of woman born can kill him (Macbeth), and that no harm will come to him until Birnam wood comes to Dunsinane. In his pride, he dismisses all of these, considering the last two impossible, thus discounting the first warning altogether. This is his third mistake: forgetting to be the humble servant he should be, even as the crowned king. He has come to see himself as invincible, a mistake in any mortal.
Thus, pushed by his ambitious wife (and all she had to do was insult his manhood) then full of his own success, he descends from the man of honor and servitude the play began with to a murderer, a man as cold and thoughtless as his wife had claimed to be. In the end, of course, Birnam wood does come to Dunsinane and he is, in fact, killed by a man who "was from his mother's womb untimely ripped," so technically, "not of a woman born."