In Act One, after Claudius asks Ros and Guil to find out what is wrong with Hamlet, Polonius appears and announces that the ambassadors from Norway are returned. He pledges his duty to the King and says he thinks he knows why Hamlet is acting mad (we later find Polonius thinks Hamlet is in love with Ophelia).
At the close of Hamlet's conversation with Ros and Guil, Polonius appears to tell him that the actors have appeared.
When Claudius and Gertrude are checking in on whether or not Ros and Guil know anything about Hamlet, Polonius buts in to announce that he is supposed to invite the King and Queen to the Tragedians' performance.
Polonius is killed off-stage by Hamlet because he was listening in while Hamlet was arguing with his mother. All we see of him after this is his body as Hamlet drags it across the stage. His death sets off the chaotic and tragic whirlwind of events that lead to the end of the play.