Was Claudius Hamlet's real father?
This is really an unusual and surprising question. At first glance, there seems to be some remote possibility that Claudius could be Hamlet's father, since Claudius and Gertrude are having such a passionate love relationship when the play opens. But Hamlet is thirty years old, and it is hard to imagine that the illicit affair could have been going on for such a long time. The only possible evidence in the play is that Claudius calls Hamlet his son, but Claudius gives no indication that he really thinks Hamlet is his son. He does not love Hamlet but only loves Gertrude. It seems far more likely that Claudius initiated the relationship with Gertrude after he had murdered his brother. And it also seems far more likely that she would have succumbed to Claudius's wooing after her husband was dead. She has strong motives for marrying Claudius other than any powerful sexual attraction. After all, as Hamlet tells her, at her age
The hey-day in the blood is tame, it's humble
And waits upon the judgement...
More than likely she married Claudius because she wanted to protect her son's right to inherit the throne and also because she would have been in a desperate position as a widow totally dependent on the goodwill of the new king. Claudius was bound to marry somebody else, and the new queen would probably hate Gertrude, might even want to have her exiled or executed. The lusty Claudius could marry a young woman and sire many princes and princesses, leaving Gertrude's son in a much worse position than he is already in.
If Gertrude and Claudius had been lovers for thirty years, then it would be logical to assume that she knew about his intention to murder his brother; but the interview between Gertrude and Hamlet in Act 3, Scene 4 makes it clear that she knew nothing.