Although Burton apparently didn't do the role the same way twice, you can see him doing something that I think it's very hard to do in performance, but that he does extraordinarily well - the same thing that Scott Shephard does in Gatz. It's a mildly metatheatrical thing: you see the actor (or the person you take to be the actor) growing before your eyes. This is a kind of anti-matter version of method acting. Hamlet isn't Hamlet (nor Gatz Gatz) at the start of the play. He's playing a role - no, he is a role, and all the sawing in the air that he does, the fake thoughtfulness, the willful passion: they're fine (as are the gestures of the actresses the narrator mistakes for La Berma when he first goes to see Phédre in Proust), and fine is dandy.
But then as the time in the theater expands, as the Shakespearean intensity of the thing builds up, our sense of the actor changes. The actor is there to act after all. But Shakespeare loves what I once called "the lost point as exile," the extended sense that now nothing will be happening, that the actors are exiled from a plot that would be going somewhere. Resignation is often an inflexion point in a story, at what screen writers sometimes call the lost point.
The orchestration of plot often takes the form of the difference between a decision weight (to use Kahneman and Tversky's terminology) and a probability. We narrate our prospects, to ourselves if to no one else, via decision weights. But we daily experience how they bump up against the "impenetrable arch of probability" (as Raymond Poincaré, cousin of Henri and Prime Minister of France during the Great War, put it when asked why he wasn't afraid he'd be killed by German bombing). Narratives bring us to this point, which is the intrusion of reality into the story. Characters and audiences have to give up our fond hopes, recalculate our decision weights, see the impossibility of what we were pulling for.
And then an efficient narrative will come to the rescue. It stymies us and then it helps us - helps us through its own unlooked for resourcefulness. (Unlooked for, or perhaps looked for but apparently not to be granted.) The end of Three Penny Opera, with Macheath's rescue from the gallows and ennoblement is a gratifying parody of this. The gratification outweighs the parody just as much as decision weights fulfilled outweigh what Truth says when she breaks in with all her realistic unlikelihoods.
But in Shakespeare, resignation lasts a long time. It's the time of resignation which is part of the time in the theater, the sense that we get of now living with these characters, rather than watching them do what they're supposed to do. We live with them in a different mode, though, from what we thought they'd be, from what they thought they'd be. Their lives are suspended, and we now belong to this interstitial time, this "interim" as Hamlet calls it, and which is to be found, variously, in Hamlet, Lear, Macbeth, As You Like It, Twelfth Night, The Merchant of Venice and, perhaps pre-eminently, in The Winter's Tale. Hamlet may be the play that thematizes this time in which time is suspended most overtly. It's a sort of play within a play, in which Hamlet himself comes to terms with the suspension that he is himself responsible for.
This occurs only when he returns from the aborted trip to England, having relinquished any thought of taking revenge on his father. True resignation in Shakespeare, true exile, comes when motives lose all power, that is to say when the ghosts disappear. The laying of ghosts is the most important purgation: Banquo ceases to haunt; Hamlet senior ceases to haunt.
Before that is the Shakespearean mess. It's this mess that his great characters tend to find so intolerable. The mess is the entanglement you can't get out of. It is, in our lives, the mess we've made of our lives: pointless, stupid, avoidable, and yet inevitable. "Just look at the filth you've made, / See what you've done" (John Ashbery, "The Task").
That's just what Burton captures so well. He's all set to play Hamlet! We see him in the court, sneering at the King, superior to his mother, all dressed in black and nobler than the image he works so hard to portray. A method actor would go deep into this opening melancholy, just as he has.
And then, to quote Dashiell Hammett, things happen. It's all a mess. He doesn't know whether the ghost is telling the truth. He doesn't know what to make of Ophelia or his love for her. Horatio is a brick, but then there's Rosencrantz and Guildenstern. He wants to confuse everyone around him, but he can do that only by being confused himself, by using his confusion as a vortex to draw others in. Why else stage a play where no one knows what's going on, who's feeling how, what emotions belong to whom, and with an added speech? It's all just a royal mess. The players come, and they seem to have some clarity, but it's false clarity - just as he's claimed from the start, actions that a man might play, but not what reality is really like. He runs into Ophelia, reading just as he's been reading for Polonius's benefit. Other people, it turns out, are just as hard to interpret and evaluate as he makes himself to them. Only his three friends, Rosencrantz, Guildenstern, and Horatio are transparent (for good or ill). And they're no help. Now he's going to stage this play, this dream of fiction, as though that could tell him the truth that the truth cannot.
So of course he starts thinking not about seeming any more but about being. The clarity of the question "To be or not to be?" is its point. What is the whole speech's attitude towards life except that it's a mess? Fardels, contumely, wrongs: where and why are they here, and why should we bear them? Being or not being are the only two clear states. And yet even they get confused: being is a mess, but dying may be just as messy, a dream of passion like the player's dream of Hecuba.
Hamlet gets clarity about the need for clarity. Burton recognizes what a mess he's in, and how pointless the mess is. That recognition isn't an achievement, the way it would be in method acting. It's a loss: the loss of motive, the loss of goal, the loss of any ontological orientation. There's nothing to be done about it. That's what the actor comes to feel, what you see Burton coming to feel. All this interaction to get through, and all of it pointless.
Until he returns from his sea change. It's at that point that he no longer wants anything. We can feel it, maybe, as the actor no longer wanting anything either. A fantastic effort has failed, but that's okay too. There is a clarity here. The role's impossible to play and that's the point.
Of course the actor plays it again every night, and that's where I think the fact that Burton played it differently every night is so significant, and (I am wagering) so captured by this specific performance (or mosaic of three performances, rather). Each night he tries to play Hamlet again. Each night the mess becomes intolerable. Each night Burton just lets the play go, lets it go where it takes him. Each night we see the education of an actor, into that place of exile on the empty stage, where everyone finally converges to acknowledge their one final achievement of straightforward truth: that this is what it's led to.