I tried to do some of the corrections myself. I think some changes must still be made. Thank you very much.

Question

1)A ghost appears to Hamlet and asks him to take revenge and kill his murderer, Claudius.
1) Correction: A ghost appears to Hamlet and asks him to avenge his murder by killing Claudius.
2)At the end of the play Hamlet manages to kill Claudius, but he also dies and his mother as well.
Correction: At the end of the play Hamlet manages to kill Cladius but in his turn dies together with his mother Gertrude.
3) Hamlet wants to be sure about what the ghost revealed him. So he arranges a play representing the Murder of Gonzago which is very similar to what had happened to his father.
3)Correction: Hamlet arranges the performance of The Murder of Gonzago whose story is very similar to what happened to Hamlet's father (to the one revealed to Hamlet by the ghost).
4) During the play Claudius raises and rushes away showing that the ghost was right.
5) This play is very important because during it the actors become the audience itself that’s why it’s called the play-within-the –play.
4)Correction: the play-within-the play is so called because the actors of the real play turn into (become?) the audience (itself).
5)Then he imagines an afterlife and he sees it as a sleep with dreams in which your troubles follow you and this is a Renaissance perspective.
6) Correction: Hamlet imagines death as a sleep. However, being a Renaissance man, he is tormented by the idea of an afterlife.