The five most important characters in Hamlet are Prince Hamlet, the Ghost of King Hamlet, Queen Gertrude, King Claudius and Laertes. Of these, Laertes is the most arguable, and many readers would certainly choose Ophelia, Polonius or Horatio instead. The textual rationale for choosing these five characters, however, is simple. They allow the central plot of Hamlet, without subplots, to take place.
Hamlet is grieving for his father's death and is also somewhat suspicious that foul play was involved. The ghost confirms this suspicion, telling him that Claudius murdered King Hamlet in order to steal his crown and marry Gertrude. Hamlet, after some prevarication, resolves to kill Claudius. Claudius, aware that his life is in danger, sets Laertes against Hamlet, and the two fight. In the final scene, Claudius, Gertrude, Laertes and Hamlet are all killed. This means that one of the most important characters was dead before the play began, and the other four die at the very end.
If one were staging a highly abbreviated version of Hamlet, therefore, these are the five characters one would require. Perhaps the weakest point is that Laertes would have no obvious quarrel with Hamlet if he had not been responsible for the deaths of Polonius and Ophelia. Claudius, however, could presumably make something up, perhaps involving these characters who could be referred to without appearing onstage. In any case, the alternative would be for the cowardly Claudius to face Hamlet himself, which seems unlikely.