I tried to do some of the corrections myself. I think some changes must still be made. Thank you very much.
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.
Great job on making some of the corrections yourself! Here are a few more changes that could be made:
1) A ghost appears to Hamlet and asks him to take revenge and kill his murderer, Claudius.
Correction: A ghost appears to Hamlet and urges him to seek revenge by killing Claudius, his murderer.
2) At the end of the play, Hamlet manages to kill Claudius, but he also dies and his mother does as well.
Correction: At the end of the play, Hamlet successfully avenges his father's death by killing Claudius, but he tragically loses his own life along with his mother, Gertrude.
3) Hamlet wants to be sure about what the ghost revealed to him. So he arranges a play that represents the Murder of Gonzago, which closely parallels his father's death.
Correction: To confirm the veracity of the ghost's message, Hamlet orchestrates the performance of The Murder of Gonzago, a play that mirrors the events surrounding his late father's murder.
4) During the play, Claudius reacts by suddenly rising and fleeing, confirming the guilt exposed by the play.
Correction: Upon witnessing the play, Claudius reacts by abruptly standing up and hastening away, affirming the truth of the accusations portrayed onstage.
5) This play is very significant because it turns the actors into the audience, thus earning the name "play-within-the-play."
Correction: The play-within-the-play earns its name because it blurs the boundary between actors and audience, as the actors themselves become the spectators.
6) Then, Hamlet envisions the afterlife as an eternal slumber, where one's troubles persist in the form of dreams, reflecting the Renaissance perspective.
Correction: Subsequently, Hamlet envisions the afterlife as an eternal sleep, where one's struggles continue to haunt them in the form of dreams—a sentiment reflective of the Renaissance belief system.