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He tells Horatio to "draw his breath in pain," to tell Hamlet's story.
Horatio
Hamlet's last few lines are "But I do prophesy the election lights on Fortinbras. He has my dying voice. So tell him, with the occurents, more or less, which have solicited. The rest is silence." This means something like, "I think Fortinbras will be elected king. He's got my vote. Tell him so, and tell him why. I can't talk any more."
Horatio lives to tell his friend's story.
Hamlet asks Horatio to recount his tale.
Hamlet asks Horatio to, "tell my story"
He asks Horatio to tell his story.
"Go, captain, from me greet the Danish king. Tell him that by his licence Fortinbras Craves the conveyance of a promised march Over his kingdom." Act 4, Scene 4
In Act five, scene one of Hamlet, we hear this conversation between Hamlet and the gravedigger: Hamlet: How long hast thou been a grave-maker? Gravedigger: Of all the days i' the year, I came to't that day our last King Hamlet overcame Fortinbras. Hamlet: How long is that since? Gravedigger: Cannot you tell that? Every fool can tell that: it was the very day young Hamlet was born. Later, the gravedigger says, "I have been sexton here, man and boy, thirty years." The conclusion is that since the gravedigger started work on Hamlet's birthday and he has been working for thirty years, Hamlet must be thirty years old.
Hamlet prevents Horatio from drinking what's left of the wine that Claudius poisoned, because Hamlet wants Horatio to tell people his side of the story.
He asked Horatio to tell others the real story of Prince Hamlet.
There is no 'narrator of the play". In some plays we see actors doing the actions while a narrator tells us what is going on. Our Town, for example, or anything written for Grade Two students. Hamlet is not that kind of play. Hamlet does have a special friend in Horatio, in who he confides, which makes him a confidant (unless he is being played as a woman, in which case she would be a confidante) and who he asks, as he dies, to "tell my story". Horatio asks Fortinbras to "let me speak to the yet unknowing world how these things came about" and Fortinbras says "Let us haste to hear it", but the play ends before Horatio narrates anything, mostly because we have seen everything which has happened and it would be boring to hear it all again. (That didn't stop Shakespeare in plays like Cymbeline or Twelfth Night, however.)