The scene begins, with dramatic economy, in medias res. The command and assent establish the employer-servant relationship immediately. We learn as we go that they are discussing Laertes.

what means: i.e., what means they have.

keep: stay, lodge.

Polonius’s approach to information-gathering is as circuitous as his speaking style.

Until I know that Hamlet has been put to death, I can never be happy no matter how lucky I am.

Only the First Folio has “were ne'er begun”; the Second through Fourth Quartos read “will nere begin” (Riverside Shakespeare). However, the rhymed version is the stronger reading, as Samuel Johnson (working from quarto texts) noted in 1765:

This being the termination of a scene, should, according to our author’s custom, be rhymed.

In a rhetorical apostrophe, Claudius addresses the King of England, saying that if he respects his command–which he should given the power of Claudius’s army–he will do what Claudius asks.

England: short for “the King of England” (see similar usage with “Norway” elsewhere in the play).

Hamlet is calling Claudius his mother and explaining this usage by saying that man and wife are one flesh; and as he is married to Hamlet’s mother, so he is of one flesh with her, and hence he is Hamlet’s mother.

Hamlet is in a manically playful state; his choice of joke may reflect his terrible ambivalence towards his mother, or may be intended merely to insult Claudius' manhood and ridicule the supposed closeness of their relationship.

Hamlet’s joke alludes to a number of biblical passages, e.g. Genesis 2:23-24:

Genesis 2:23: And Adam said, This is now bone of my bones, and flesh of my flesh: she shall be called Woman, because she was taken out of Man.

Genesis 2:24: Therefore shall a man leave his father and his mother, and shall cleave unto his wife: and they shall be one flesh.

See also Matthew 19:4-6.

Claudius’s “purposes” are to have the King of England execute Hamlet on his arrival.

This is an apparent slip or warning on Claudius’s part, hinting at his ill intentions towards Hamlet. Claudius may be too drunk or infuriated to think clearly, or may simply be overconfident in his power.

If we credit Hamlet’s later account of the events aboard ship that led to his discovery of Claudius’s plot, the Prince is not yet fully or consciously aware of the king’s “purposes.” However, his retort indicates that he smells a rat.

These lines indicate that Hamlet’s mind, though vast in intelligence and memory, does have limits, as he struggles to recall the exact lines. Impressively, he corrects himself and does deliver the speech purely off memory, from a play he likely witnessed years ago.

Hamlet’s mini-performance is often played as being genuinely skillful, since he’s a lover of the theater. The 2009 David Tennant version (taking its cue from this initial stumbling) makes the choice of having Hamlet look rather amateurish:

https://youtu.be/FnrKScWFR3M?t=45s

But it was–as I took it, and others whose judgment in such matter drowns out mine–an excellent play, with well-ordered scenes that were clever but not overdrawn.

A model perhaps Shakespeare kept in mind for his own plays.

cried…of: “were louder than, i.e. carried more authority than” [Riverside Shakespeare].

Hercules' load is the globe, a subtle reference to Shakespeare’s Globe theater. The Riverside Shakespeare notes:

Hercules in the course of one of his twelve labors supported the world for Atlas; the children do better, for they carry away the world and Hercules as well. There is an allusion to the Globe playhouse, which reportedly had for its sign the figure of Hercules upholding the world.

Shakespeare is using this section of Hamlet to mock the fashions of the London theater, which at that time favored the dumb-shows of cute children. “Carry it away” suggests that they have successfully swept away the kind of dense, intelligent drama that was Shakespeare’s company’s forte. However, there is no evidence that Shakespeare’s company suffered much from the competition, and his audience may well have delighted in the flat contradiction of a packed house.

Throughout this verse, Trainor slams this boy for acting differently behind her back. She’s tired of his disrespectful behavior, and warns him not to come crying to her now that they’re over.

Take note of Trainor’s pun when she twists the word “too”:

Hey baby don’t you bring them tears
Cause it’s too late, too late baby
You only love me when you’re here
You’re so two-faced, two-faced babe