What key moments would have had to have been altered in order for a positive outcome to be created for the characters within Hamlet?

This, of course, depends on what we mean by a positive outcome. If we take that to mean that Claudius alone should have paid for the crime of murdering Hamlet's father, a positive outcome would be Claudius's death in the absence of all the other corpses littering the the stage at the end of the play.
One way this could have happened is the following: once Hamlet had confirmed throughout the Mousetrap play that Claudius was truly a murderer, he could have immediately gone, with several reliable witnesses, and forced Claudius to confess to his crime. Then, Hamlet could either have killed Claudius or, even better, brought him before a court or tribunal. That way, Hamlet would not be taking the law into his own hands.
Either way, the key point is that to have a good outcome, it would be best to avoid having Hamlet mistakenly murder Polonius. Polonius's death, especially with Claudius still in power and able to manipulate Laertes, set off the the chain of events that led to the final tragic ending.