Enter King and Laertes:
As the scene opens, the King has just about finished the job of making Laertes see things his way. Not only did Hamlet kill Polonius, but "he which hath your noble father slain / Pursued my life" (4.7.4-5). Laertes has just one more question: Why didn't the King bring Hamlet to account for the murder?
The King has two reasons. The first is that he loves Gertrude, and Gertrude loves her son. She "Lives almost by his [Hamlet's] looks" (4.7.12). The second reason is that Hamlet has the love of "the general gender" (4.7.18); the King means that everybody loves Hamlet, and so any accusations against him might backfire.
It's hard to tell how much truth there is in the King's explanations, given his need to cover himself, but he's probably not telling absolute lies. Gertrude is indeed protective of Hamlet, and the fact that a crowd shouted "Laertes shall be king!" (4.5.107), seems to show that the King has reason to fear the "general gender." Still, it seems to me that the primary reason the King didn't hold Hamlet accountable for the killing of Polonius is that the King strongly suspects that Hamlet knows who killed King Hamlet. If that's the case, the last thing the King wants is a public confrontation with Hamlet; having Hamlet killed out of view, in England, is much more suited to the King's way of doing things.
Laertes buys the King's explanations and vows revenge against Hamlet. The King reassures Laertes that he needn't lose any sleep over that. He, the King, is not somebody to be messed with, and he loved Polonius.
If Laertes had indeed confronted Hamlet, that would have been a manly thing to do, but the King talks him into doing a cowardly thing. He tells Laertes he has a plan to kill Hamlet, one so cunning that "for his death no wind of blame shall breathe" (4.7.66). Laertes is all for it, especially if he can be the one who actually kills Hamlet. Still, the King seems to have some doubts about Laertes' willpower, because he gives Laertes a thorough psychological working-over before he actually reveals the plan.
The King starts with flattery, telling Laertes that Hamlet envies him "for a quality / Wherein, they say, you shine" (4.7.72-73). Laertes asks what that quality is, but the King leads him on, telling him that the quality is "A very riband in the cap of youth" (4.7.77). Then the King starts talking about a certain gentleman of Normandy, and what a wonderful horseman he is. The idea seems to be that anyone would be proud to know this gentleman, and Laertes gets sucked in. He identifies the gentleman as "Lamond," and says, "I know him well: he is the brooch indeed / And gem of all the nation" (4.7.94-95). Now, when he has Laertes all excited about Lamond, the King says that Lamond has praised Laertes "For art and exercise in your defence / And for your rapier most especial" (4.7.97-98). And this is why, according to the King, Hamlet envies Laertes.
"Now, out of this--," says the King, as though he's about to lay out his plan, but when Laertes asks, "What out this, my lord?" the King asks a question that is very close to being an insult: "Laertes, was your father dear to you? / Or are you like the painting of a sorrow, / A face without a heart?" (4.7.107-109). The King goes on to explain that he doesn't doubt that Laertes loved his father, but love itself can die. He says, "I know love is begun by time; / And that I see, in passages of proof, / Time qualifies the spark and fire of it" (4.7.111-113). From this, the King draws the lesson whatever we're going to do, we should do right away: "That we would do / We should do when we would" (4.7.118-119). Otherwise, the longer we delay, the less likely we are to act, until the thing we said we would do becomes only something we should do, sometime or another.
The King has put enormous pressure on Laertes. First he flattered him, and then he challenged him to prove that he really loved his father. Naturally, when he asks Laertes what he would do "To show yourself indeed your father's son / More than in words," Laertes answers "To cut his throat i' the church" (4.7.126).
This exchange between the King and Laertes has echoes from many other places in the play. The King's suggestion that Laertes may be "a face without a heart" reminds us of what Hamlet said of his sorrow for his father, "I have that within which passeth show" (1.2.85). The King's assertion that "time qualifies the spark and fire of love" reminds us that when the Ghost appears in his mother's closet, Hamlet described himself as "lapsed in time and passion" (3.4.107). Also, the player King said "'tis not strange / That even our loves should with our fortunes change" (3.2.200-201). And when the King tells Laertes that what "we would do / We should when we would" we're reminded of Hamlet's struggles with the question of "Why yet I live to say 'This thing's to do'" (4.4.44). Finally, Laertes' assertion that he would "cut his throat i' the church" reminds us that Hamlet had the chance to kill the King while he was praying. Hamlet even got so far as saying, "And now I'll do't" (3.3.74), but he didn't.
Returning to the King and Laertes, we see that only now, when he has Laertes thoroughly hooked, does the King reveal his plan. When Hamlet comes home Laertes is to stay away from him, but the King will get people to praise Laertes' skill with his rapier in front of Hamlet, so that Hamlet will become even more envious. (In the following scenes we don't see or hear anything more of this first part of the King's plan.) Then a wager will be made that Hamlet can't best Laertes in a fencing match. At the fencing match Laertes will have a "sword unbated," that is, one without the protective button on the sharp end. At this point, the King gives Hamlet an unconscious compliment, saying that "He, being remiss, / Most generous and free from all contriving, / Will not peruse the foils" (4.7.134-136). So Laertes will be able to get his revenge accidentally on purpose.
Laertes not only agrees, he gets into it. He has a poison that he can put on his sword, so that a mere scratch will kill Hamlet. (Are we supposed to ask what Laertes is doing with that poison in the first place?) Laertes' back-up plan inspires the King to come up with a back-up to the back-up. He thinks things through, and worries that their plans might fail, then has a bright idea. Laertes is to make the fencing match very vigorous, so that they will get "hot and dry." Hamlet will call for drink, and the King will give him chalice of poisoned wine. That ought to do it.
At the news, Laertes tries to fight off tears, but without success. First he makes a kind of sad joke, saying "Too much of water hast thou, poor Ophelia, / And therefore I forbid my tears" (4.7.185-186). But the tears come, and Laertes promises that "when these are gone, / The woman will be out" (4.7.188-189). That is, when he has finished crying, he will have gotten all of the "woman" out of himself and be a man again. He exits, still weeping, with his best-remembered lines: "I have a speech of fire, that fain would blaze, / But that this folly drowns it" (4.7.190-191).
The King, true to form, is worried about his own skin. He tells the Queen that they have to follow Laertes because the news of Ophelia's death may ignite Laertes' rage again. So it seems that the King, even after all of his manipulation of Laertes, is still afraid of him.