Do think the play 'Hamlet' should have ended at his death or do you feel that that the next 50ish lines that follow are important and critical? Why or why not?
Ending the play at Hamlet's death would leave several practical plot points unresolved and the audience would also be deprived of some crucial character development.
In the last lines of the play, Hamlet makes a pair of dying wishes: he tells Horatio to spread the news of what really happened, and he asks that Fortinbras be named king. The play, Hamlet, then seems to be Horatio's touching fulfillment of Hamlet's request. Hamlet's wish that Fortinbras be made king showcases how highly Hamlet thinks of this prince he has never even met: recall that Hamlet, in his "How all occasions do inform against me" speech in Act 4, Scene 4, praises Fortinbras's ability to move boldly even on seemingly trivial matters.
However, without the last few lines that come after Hamlet's death, we would not see how truly devoted Horatio is to his noble-blooded friend.
Shakespeare wrote in the early modern period (the late 16th and early 17th century). During this time, in the archaic rules of the English language there were two different ways to use a second person singular pronoun: you and thou. The difference between these two words was actually significant: you used "thou" when you were talking to social inferiors or to people you were familiar with or if you wanted to show scorn ("Get thee to a nunnery!"), and "you" to social superiors and when you were being formal.
Up until this point in the play, Horatio has used the formal "you" whenever he is referring to Hamlet. Now that Hamlet has passed away, Horatio switches to the more familiar "thou," revealing how close his relationship had been with Hamlet.
No, I do not think that the play should have ended with Hamlet's death.
First, as Hamlet is dying, he tells Horatio to not commit suicide. Hamlet wants Horatio to stay alive, so that he can tell everybody the truth of what has transpired. It's Hamlet's dying wish that Horatio explain the events. It would be disappointing to an audience to not know if Horatio carries out Hamlet's last wish. There would always be a "loose end" hanging over the ending to the play. Horatio does indeed tell the events to Fortinbras thus fulfilling Hamlet's last wish.
The other part of Hamlet's last words to Horatio is that Fortinbras be made king. Without that last 50 or so lines, the audience wouldn't know if Fortinbras is indeed planning on taking over the throne. Those two reasons, Horatio telling the story and Fortinbras agreeing to the throne, are reason enough for Shakespeare to include those lines of text for the audience.
The final 9 lines of the play are necessary, because they show the audience that Fortinbras is treating Hamlet with a great deal of respect. He wants his body carried out by four captains and given all of the dignity of a soldier's death.
Shakespeare fittingly ends Hamlet by having Horatio fulfill the prince's last wishes and depicts Fortinbras's election as the new King of Denmark. Being that the play's noble protagonist's final wishes are for his close friend to tell his story and for Fortinbras to inherit the throne, it is fitting that Hamlet receives his last requests. Horatio's last words to Prince Hamlet are also significant and suggest the prince's final resting place. Horatio says,
Now cracks a noble heart.—Good night, sweet prince, And flights of angels sing thee to thy rest! (Shakespeare, 5.2.358-360)
The image of angels singing as they fly while Hamlet rests implies that the prince is in heaven, which justifies his actions and leaves the audience with a hopeful sentiment following his tragic death. Horatio displaying the corpses to the public and telling Prince Hamlet's story concludes the tragedy while Fortinbras's ascension to the throne resolves the issue of having a power vacuum in Denmark. Fortinbras's treatment of Hamlet's corpse and his kind words end the play on a lighter note, which creates an aesthetically pleasing conclusion to the tragedy.