In Shakespeare's Hamlet, exactly what causes the death of each of these characters: Gertrude, Claudius, Laertes, Hamlet?
In Shakespeare's Hamlet, the deaths of all of the characters you mention are the result of Claudius' murderous conniving.
Claudius talks Laertes into killing Hamlet to avenge Polonius' death. They arrange for a friendly game of "sword play," but Laertes poisons the tip of one sword as part of the their plan to dispose of Hamlet.
As a backup plan, Claudius also has wine at the event, and allegedly drops a large pearl in the cup to show his faith in Hamlet's abilities to win, saying he will have the pearl as a reward: the pearl, however, is also poisonous.
Stay, give me drink. Hamlet, this pearl is thine;
Here's to thy health.
Drum, trumpets, and shots. A piece goes off.
Give him the cup. (V.ii.283-285)
As the sword play begins, Hamlet acts lighthearted, even though he expects treachery from some source. Laertes loses his sense of "play" and becomes aggressive with Hamlet, drawing blood with the poisoned sword.
In a scuffle, the swords are exchanged and Hamlet, unknowingly, ends up with the envenomed weapon. Hamlet then cuts Laertes with the same poison that was meant for him.
Meanwhile, Hamlet has had no interest in drinking, but Gertrude raises the poisoned cup in honor of her son, and drinks. Claudius could have stopped her, but does not.
Gertrude stumbles and falls, gasping that she has been poisoned:
Gertrude, do not drink.
I will, my lord; I pray you pardon me. (V.ii.294-295)
Gertrude dies. Hamlet wants the room sealed to find who is responsible for the treachery. Laertes falls and admits it rests with him.
No, no! the drink, the drink!—O my dear Hamlet!—
The drink, the drink! I am poison'd. (V.ii.317-318)
It is here, Hamlet. Hamlet, thou art slain;
No medicine in the world can do thee good.
In thee there is not half an hour of life.
The treacherous instrument is in thy hand,
Unbated and envenom'd. The foul practice (325)
Hath turn'd itself on me. Lo, here I lie,
Never to rise again. Thy mother's poison'd. (V.ii.321-327)
Before Laertes dies, he points to the King, saying that Claudius arranged it all.
The King, the King's to blame. (line 328)
Hamlet goes to Claudius and stabs him with the poisoned sword and forces him also to drink the poisoned wine.
The point envenom'd too! Then, venom, to thy work. (line 329)
[Hamlet stabs the King.]
Hamlet returns to Laertes who asks for Hamlet's forgiveness, which he grants. Laertes dies, and soon, so does Hamlet—after asking Horatio to tell the story of what has happened, and giving over the kingdom to Fortinbras who has just won Poland.
Here, thou incestuous, murderous, damned Dane,
Drink off this potion! Is thy union here?
Follow my mother. (lines 332-334)
The play ends with Horatio's famous lines:
Good night, sweet prince,
And flights of angels sing thee to thy rest! (lines 371-372)
Because Shakepeare's play Hamlet is a tragedy, there are numerous deaths in the final scenes. Also, according to Aristotle's definition of tragedy, these deaths are brought about because of the protagonist's (Hamlet) tragic flaw. In this play, Hamlet's tragic flaw is that of inaction. His mother Gertrude meets her untimely demise by toasting to her son. To her surprise, the cup of wine she held in her hand was poisoned and was meant for Hamlet. Therefore, plan A was to poison Hamlet during the duel if he seemed to be winning. In addition, they also poisoned the Lartes' foil (sword) with a poison so powerful that even a scratch would kill Hamlet. However, when grappling with Laertes, the two accidentally switched swords so that Hamlet was left holding the deadly weapon. In the fight, both men were exposed to the poison and Laertes confessed the King's plan to Hamlet. In a bout of rage, knowing that the tip of his foil was dipped in poison, Hamlet stabbed Claudius - killing him in the process.