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.

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KING:

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.

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KING:

Gertrude,
do not
drink.

QUEEN:

I
will, my lord; I pray you pardon me.
(V.ii.294-295)

Gertrude stumbles and
falls, gasping that she has been poisoned:
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QUEEN:

No,
no! the drink, the drink!—O my dear Hamlet!—

The drink, the
drink! I am poison'd.
(V.ii.317-318)

Gertrude dies. Hamlet
wants the room sealed to find who is responsible for the treachery. Laertes falls and
admits it rests with him.
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LAERTES:

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.

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HAMLET:

The
point envenom'd too! Then, venom, to thy work. (line
329)

[Hamlet
stabs the King.]

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HAMLET:

Here,
thou incestuous, murderous, damned Dane,

Drink off this
potion! Is thy union here?

Follow my mother. (lines
332-334)

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.

The play ends with Horatio's
famous lines:

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Good night, sweet prince,
And flights
of angels sing thee to thy rest! (lines
371-372)