At one point, Hamlet questions his stated mission, suspecting that his father's ghost, who has set him upon the road to revenge, may not be telling the truth. "The spirit that I have seen/ May be the devil; and the devil hath power/ To assume a pleasing shape; yea, and perhaps/ Out of my weakness and my melancholy,/ As he is very potent with such spirits,/ Abuses me to damn me." What if his uncle is innocent of murder? If Hamlet kills him, would he not then be guilty himself, and have condemned himself to hell?
Or what if, instead of being guilty of murder, his uncle did kill his brother, but had a justified reason to do so? What if, for that matter, Hamlet's actual father isn't dead, and the Ghost is urging Hamlet to kill him?
I suspect it's rare to find an actor who's played Claudius who didn't at least speculate on who Hamlet's father really is. How quickly did Claudius and Gertrude marry? Is Claudius so spectacularly sexy and charismatic that his brother's widow is unable to resist him even within a few weeks of a beloved husband's death? Or is this a long-standing affair, and Hamlet's perception of his parents' happy marriage is simply wrong?
People have been suggesting that Claudius is Hamlet's real father even longer than they've been suggesting that Shakespeare wasn't really Shakespeare. Questions about Hamlet's paternity may well have first come up sometime in the middle of the first performance. Questioning Shakespeare's authorship had to wait for all the witnesses to die, a couple hundred years to pass, and Delia Bacon to go nuts and suggest the Bard was too uneducated to write that well.
Consider the way Claudius treats Hamlet early in the play. He appears to be making every effort to be a good step-father. The advice he gives him is sound. He wants Hamlet to remain at Elsinore, rather than return to school, as keeping him there will please his mother. If he held any initial enmity toward his step-son, it would have been far more logical to hurry him on his way. Is he so madly in love with his new bride that he's willing to risk keeping a dangerous rival nearby? Or does he genuinely care for the young man, presuming him to simply be depressed by his father's death and figuring he'll come out of it in time, just as most people do?
His reaction to Hamlet's apparent madness is at first fairly sensible and generally benign. He has Rosencrantz and Guildenstern spy on Hamlet, and report back. Is he genuinely mad? If he is, does he seem dangerous? It's not until he joins Polonius in spying on Hamlet and Ophelia that he begins to seriously question his step-son's motives, and decides to send him away to England. And note, at this point the intent seems to be that and nothing more. The decision to have England kill Hamlet doesn't come until after Hamlet has killed Polonius, and made it blindingly clear that he only did so because he thought he was killing Claudius.
In this sort of situation, no matter how much he might care for his step-son and for his wife, he's not going to keep such a dangerous young man within striking distance. He'll get him away, and find a way to eliminate him that won't alienate Gertrude. He still loves his wife. Moreover, despite the drama in the queen's bedchamber, she appears to still love her husband. She may have gone along with Hamlet's advice while the prince was still there with his sword dripping Polonius' blood, but once he's gone she's back in her husband's bed.
Really, if you were Gertrude, and your son came to you with this story, would you believe him? Particularly if he started talking to an empty doorway in the middle of his harangue. She was humoring him because now she was sure he was mad, and dangerous, not because she believed him.
Or, perhaps, she did believe him, not because of what he had to say, but because she'd been in on the murder from the beginning. As I said earlier, her haste to remarry is certainly suggestive of a long-term affair. Hamlet is sure his parents had an ideal, loving marriage, but most children think that. Sometimes the only thing they have is a talent for presenting a happy facade.
The Ghost argues in favor of a happy marriage, but he would, wouldn't he? He wants revenge. He refers to his brother as "that incestuous, that adulterate beast." The incestuous part comes from the weird biblical declaration that having sex with a brother's wife is incest, which, honestly, it obviously isn't. There's an exception if the first marriage was childless, but that clearly wouldn't apply here. Shakespeare includes it, and Hamlet holds to it, just as strongly as does his father, and that's what matters for the play.
Claudius and Gertrude, equally, do not hold with that belief. Apparently, neither did the priest who married them. It's an Old Testament dictate, so perhaps they argued it didn't apply to Christians (though, with the usual dating of the play, Denmark wouldn't have been Christian yet anyway, and pagans wouldn't have even cared about an ancient Jewish law).
But the important part isn't the incest. The important word here is "adulterate." The Ghost is flatly stating that his wife was screwing his brother while he was still alive. He's confirming that this was an existing affair. He may even suspect, or know, that it dates back many years, and that Hamlet may not actually be his son. Think of that in terms of revenge!
By having Hamlet kill Claudius, who is his real father, the Ghost gets his revenge not only for his own death, but for everything that went before. His brother is killed by his own son, even if the son is unaware of this, sending his brother to hell for King Hamlet's murder, and Hamlet to hell for Claudius' murder. Gertrude, who loves her current husband and has for many years, will be devastated by the knowledge that her son has killed not only the man she loves, but the man who fathered him. Murder is bad, patricide is worse. Maybe, if he's lucky, Gertrude will become so depressed she'll kill herself and end up in hell with the other two.
Potentially, this could tie in with why Claudius, and not Hamlet, became king. Perhaps his father did suspect he wasn't his real father. Shakespeare never says why the council picked Claudius. If old King Hamlet had secretly declared his son a bastard, that would certainly drop him from the line of succession just as the marriage of his putative real father and his mother would legitimize him and put him back into it.
Of course, the reason could just as easily be Hamlet's seeming perpetual student status. Wittenberg is a long way from Elsinore, so just notifying Hamlet that his father had died, and giving him time to return, would take a month or more. Perhaps it was thought that, with the threat from Fortinbras, there wasn't time to wait.
It's always interesting to speculate on these things. When I played Claudius in 2015, we made quite a few cuts. There was no scene one, Rosencrantz and Guildenstern were eliminated (along with about 45 minutes of tedious dialogue), and the entire Fortinbras subplot was cut out. It made for a good deal more action and a good bit less talking. And you can do this to Shakespeare because he's been dead for 400 years and the plays are all firmly in the public domain.
In 2016, I wrote a Hamlet prequel, To Kill a King, that employed many of these speculations. Hamlet very likely is Claudius' son, and Gertrude has been in love with Claudius ever since the day she first saw him and realized she was going to have to marry his brother. There's no long-term affair, just a drunken slip 30 years earlier, but she's fairly sure that night was when she conceived.
King Hamlet and Gertrude don't get along, but seem to be happy whenever anyone is around. In private, they cordially hate each other. King Hamlet loves war far more than he loves his wife or his son. He does, however, seem inordinately fond of Osrick.
Claudius kills his brother, but only because his brother sent an assassin to murder Gertrude. And only after a lot of soul-searching, for he has to justify both fratricide and regicide, and at best can only conclude that if it save's Gertrude's life, even hell is worth it.
In effect, in killing his brother, Claudius is simply returning the poison intended for Gertrude to its supplier. And, from that point, with a dying King Hamlet swearing revenge even from beyond the grave, and Hamlet upset, moody, and depressed, and in complete ignorance of his uncle's conspiring with the council to legitimize his now secretly-acknowledged son and allow him to succeed him, things will carry on as they always have. The last words of my longish one-act play are the first words Claudius says in Shakespeare's.
Really, did anyone think actors just read the words without bouncing off every motivational tangent on the way to the performance? I'm certainly not a "method" actor, but I do like to have some idea why a character is doing these things.