Forestall…hither: delay their coming here.

Horatio knows that Hamlet suspects the swordfight is a plot against him, and offers to forestall it. Hamlet, for once, rejects the delay.

“Fit” means “athletically fit,” but recall Hamlet’s description of Claudius in 3.3 as “fit and season’d for his passage” into death.

gain-giving: misgiving.

This misogynistic opposition between supposed martial virtue and female foolishness or frailty is repeated in various contexts throughout the play. See 1.2: “Frailty, thy name is woman!”

the noblest: i.e., the highest-ranking nobility.

Like “stage,” “audience” ends the play on an appropriately meta-theatrical note. An audience is to gather around a stage bearing the bodies of the dead, where Hamlet in particular is to be celebrated.

Cousin: then a general term of address for a relative.

Riverside Shakespeare notes (2nd ed., p. 1232):

laid the odds: i.e. wagered a higher stake (horses to rapiers).

is better’d: has perfected his skill.

odds: i.e. the arrangement that Laertes must take more bouts than Hamlet to win.

The exchange is full of irony: the King “do[es] not fear” losing his literal wager on Hamlet, because he’s arranged for Hamlet’s death. But as noted above, he’s metaphorically gambled everything on the outcome of his scheme.

The only judge we see in the scene is Osric. He may or may not be in on the plot; see note above.

Either way, this may be another subtle Shakespearean joke: judgment in Claudius’s Denmark is about as fair, competent, and perceptive as Osric. See Claudius in 3:3:

In the corrupted currents of this world
Offence’s gilded hand may shove by justice…

“Judgment” is one of the play’s major themes: see e.g. “accidental judgments” below, “whose blood and judgment are so well commingled” in 3.2, etc.

There is a running motif in Hamlet of schemes blowing up in the schemer’s face, characters being caught in their own traps, and so on. Compare:

For ‘tis the sport to have the engineer
Hoist with his own petard…

Ultimately this connects with the larger theme of fate overthrowing our purposes, as expressed by the Player King (reciting Hamlet’s words?) in 3.2:

Our wills and fates do so contrary run
That our devices still are overthrown;
Our thoughts are ours, their ends none of our own.

Grimly ironic. Mention of succession reminds us that Claudius has stolen the Danish crown–and all its accompanying riches–from both Hamlet and his father. Now he intends to ensure that Hamlet will never succeed him.

Danish Crown Regalia. Via Wikimedia

kettle: kettledrum.

cannoneer: soldier in charge of the cannons.

Compare from 1:4:

The king doth wake to-night and takes his rouse,
Keeps wassail, and the swaggering up-spring reels;
And, as he drains his draughts of Rhenish down,
The kettle-drum and trumpet thus bray out
The triumph of his pledge.

The king’s mode of “celebration” is a reminder of his debaucherous parties and general corruption. It also sets up the “warlike noise” of artillery fire later in the scene, heralding the arrival of Fortinbras, and the gun salute for Hamlet at the end of the play. These sounds converge and take on multiple layers of symbolism: violence, excess, celebration, death.

The king promises to celebrate any of several feats by Hamlet with a cannon salute from the castle battlements, as well as a toast to his “better breath” (health, superior athleticism). After drinking from the cup of wine, he’ll throw in a union–rare pearl–as an additional reward.

The king means to show that the wine is safe to drink before dropping in the pearl: the agent of poisoning.

Hamlet puns grimly on “union” toward the end of the scene.

Cannon at the French Spur, Stirling Castle, Scotland. Via Wikimedia

Riverside Shakespeare notes (2nd ed., p. 1232):

stoups: tankards.

quit…exchange: pays back wins by Laertes in the first and second bouts by taking the third.

A stoup of wine also figures in another evil Shakespearean plot: Iago’s scheme to get the lightweight Cassio drunk in Othello 2.3.