How has poison killed the characters of Hamlet?

None of these characters is actually killed by poison. Polonius is stabbed by accident -- Hamlet thinks he's Claudius. Ohelia drowns herself. Rosenkranz and Guildenstern are supposed to take Hamlet to England where Claudius has instructed the English king to kill him, but Hamlet finds out and changes the letter. The poison is the hatred and paranoia in the minds of the characters which leads to the deaths.