Poison is present throughout Hamlet. The first mention of poison in the play comes when old, dead king Hamlet visits his son as a ghost. During this scene, ghost Hamlet tells young, living Hamlet that Claudius (old Hamlet's brother) killed him by putting poison in his ear.
The second occurrence of poison occurs when Hamlet plots to have the traveling actors perform a play in which a fictional king is killed with poison in a very similar manner to how King Hamlet was killed. Prince Hamlet is hoping to watch Claudius's reaction to confirm that he is guilty.
Poison shows up yet again when Laertes vows to kill Hamlet for killing his dad accidentally. He challenges Hamlet to a sword fight and puts poison on the blade. To make doubly sure that Hamlet dies, Claudius poisons the wine that Hamlet is likely to drink from.
Plans don't quite work out from here. Hamlet declines the wine, but Gertrude unknowingly drinks from it and dies. Laertes does wound Hamlet with the poisoned blade, but Hamlet survives long enough to disarm Laertes and stab him with his own poisoned sword. Then Hamlet stabs Claudius with the poisoned blade AND forces him to drink the rest of the poisoned wine. Many people die.