He has the chance to, just after the Mousetrap play. He decides not to because he worries for some reason that Claudius will go to heaven if he dies praying. Hamlet wants him to go to Hell.
Hamlet mistakenly stabs Polonius through the curtain. He heard a noise, and, thinking it might be Claudius, immediately stabbed through the curtain.
Claudius hopes that Hamlet will be killed in England. Claudius is the King and is also the brother of Hamlet.
Before Hamlet enters his mother's chamber, Polonius is having a discussion with Hamlet's mother Gertrude. As Hamlet enters the room, Polonius conceals himself behind a tapestry hanging from the wall. When he hears Gertrude begin to feel threatened by Hamlet, he cries out for help, thinking that he would kill them both. Hamlet hears Polonius, and believes him to be a rat/spy (possibly even King Claudius, but it is unclear if he truly believes this). Hamlet stabs through the tapestry killing Polonius, who remarks "O, I am slain".
His brother, Claudius, came out into the garden while King Hamlet was sleeping and poured poison into his ear.
Claudius is married to Gertrude (Hamlet's mother and widow to the former king).
His mother Gertrude called for help and a voice came from behind the curtain repeating the call for help. Hamlet assumed that it was Claudius and his chance to kill him had come so he stabbed through the curtain. It was only after that he found that it wasn't Claudius after all.
Gertrude, Hamlet's mother and wife of his Uncle Claudius, King of Denmark.
Hamlet's uncle Claudius killed Hamlet's father (called Hamlet Sr.). The ghost of Hamlet Sr. comes back from the dead and tells Hamlet Jr. so. Hamlet Jr. confirms this by re-enacting the murder as a play, which upsets Uncle Claudius.
No, after the dual. Claudius lets Hamlet's mother die, and Hamlet and Lateres both wound each other, than Hamlet forces the poison down Claudius' throat.
Claudius sends Hamlet to England, but he doesn't get there.
Hamlet's father is dead. His uncle (Claudius) marries his mother. Hamlet sees his father's ghost and the ghost tells him that Claudius murdered him. Hamlet seeks revenge and kills Claudius. Hamlet also dies. There are many others plots in Hamlet but that is the main one
Hamlet finds Claudius praying for forgiveness after the play. He decides that if he were to kill Claudius at that moment, Claudius would go to heaven rather than to hell. Hamlet decides to wait until he finds Claudius sinning, in order to kill him.