The sequence doesn't start with the usual rhymes in this version, but instead with Horatio trying to tell Hamlet that was he noted most definitely was NOT conclusive proof of Claudius' guilt. Hamlet won't listen, of course, an almost violent euphoria overtaking him. He asks for music, but also grabs a silver tray from a servant, banging it in the man's face like a gong. Horatio attends him and laughs at his jests, but he doesn't speak or try to broach the subject of the King again. The focus changes, in any case, to Rosencrantz & Guildenstern, surely Horatio's rivals in the friendship department, so perhaps there's a psychological, personal reason why he falls in with Hamlet again despite the unspoken disagreement.
David Tennant's Hamlet is at his funniest in this scene. His "with drink?" is impish and a confidential "he's drunk again eh? yeah, that happens a lot", traitor's talk that makes R&G co-conspirators if anyone were listening. As they continue, Hamlet more or less ignores them, putting the Player's crown on his head, using the silver tray as a mirror, hiding his face from them. He's their mock King, the King that should have been if only they (and the rest of the Court) would follow the right man, the true heir. He's also mocking Claudius, sitting on the throne misunderstanding everything they say on purpose. It is a sly condemnation of Claudius' decisions to date, prefiguring Denmark's downfall at the hands of Fortinbras. When they demand an answer from him, he raises his hand excitedly, like a school boy, only to reveal he can't make a wholesome one. It's with mock pride that he says the "oh wonderful son" line. It's all a caricature of his false King/Father.
And he does the same with the friendship he bears these men. At "pickers and feelers", he gets up to tickle Rosencrantz, freaking him out. And he should. They are failing at their appointed task, and that will have consequences. For this Rosencrantz, it's really about getting it over with so he can leave this hellish place before the axe falls on him. He tries to keep back, he's rather sincere when he asks Hamlet to stop what he's doing before it ruins them all. Guildenstern is the more political, albeit gauche, animal. He tries to find the right words and hopes to get favor from the King. But of course, neither of them are aware that Claudius may be a murderer and usurper. They just don't get it, perhaps because they can't. It's shown, for example, when Rosencrantz gives Hamlet a blank stare at "While the grass grows". There's a funny beat before the next line, and Hamlet obnoxiously slaps him, but doesn't awaken him to the very real danger of trusting Claudius.
The recorders arrive, and the production plays on the plural in that line to make Hamlet throw one to Horatio, which will come into play later. The interplay between Hamlet and Guildenstern is sweeter and less violent than usual, initially, Hamlet acting all innocent, not understanding why Guildenstern won't play and laughing at the irony. Guil, for his part, tries to understand what is being said and takes the Prince's explanations with a measure of respect, as if he agrees he's been rumbled. Then BAM! A rare musical sting accompanies Hamlet attacking Guildenstern with the pipe, trying to choke him, even as Rosencrantz tries to gently separate them. What actually does it is Polonius' arrival.
Coming into a violent moment, Polonius gets a loud pipeful in the face, followed by Hamlet and Horatio playing a tune together over his words. By having the old councilor mouth words at R&G with a certain degree of ire, the production motivates his presence. Thematically, I've said that Polonius is entirely redundant in the scene because he always makes the wrong choices, and that Hamlet feels he's irrelevant all the time. Here, he comes because R&G are taking too long. We have to remember he plans to hide in the Queen's closet (even if he hasn't said so yet), so his impatience stems from that. He's been kept waiting as much as the Queen has. Showing impatience and even anger is rare for him, so perhaps his head too, is on the chopping block. When the King is angry, chaos reigns in Elsinore. And it's because he's impatient and in a hurry that Hamlet chooses to play the game of clouds with him, pushing him to show disrespect. Polonius catches himself each time, but his composure does break.
The soliloquy is done straight into Hamlet's movie camera, a point of view we get to see. Sitting askew on the throne while he speaks of being cruel to his mother creates a visual inversion of what is natural, and may put the lie to his words. Hamlet looks quite demented here. His successive victories over Claudius, R&G, and Polonius have pushed him over the line of what may be reasonable or acceptable.