Initially Hamlet seems to think that Gertrude was having an illicit relationship with Claudius before he murdered his brother and that she was in on the plan. In Act 3, Scene 4, however, he seems to realize that she had no idea that Claudius had killed her husband. When Hamlet kills Polonius, she says, "O, what a rash and bloody deed is this!" Hamlet replies, "A bloody deed? Almost as bad, good mother, / As kill a king, and marry with his brother." Gertrude seems genuinely astonished when she says, "As kill a king!" It is hard to understand exactly what happened between Gertrude and Claudius. It is never made explicit. It would seem that Claudius had no kind of understanding with her before he killed King Hamlet but that he wooed her after he had committed the murder. This also seems to be Hamlet's eventual understanding of what happened.
Hamlet does wonder the extent of Gertrude's involvement. People overlook that throughout the play, Hamlet is as upset if not more so with his mother than Claudius. The first scene in The Mousetrap shows that Hamlet is not going to let his mother off easy as the ghost had directed. The Mousetrap is called The Mousetrap because it is to catch Gertrude. Only the second scene known as The Murder of Gonzago is pointed at Claudius.
In this first scene the Player Queen makes pointed protest about killing husbands. In Jenkins' Arden Hamlet, Jenkins notes that the Player Queen isn't suggesting she actually murder her first husband but that her actions encouraged it, thereby making her complicit. The aim is to test whether Gertrude has any guilt to unkennel with regard to King Hamlet's death. She does pass the test and makes no connection between her first husband's death and Claudius's interest in her.
Another interesting exchange though seems to indicate that Gertrude is clueless about Claudius's skulduggery. In Act 2 Scene 2 after Polonius is sent to receive the Ambassadors from Norway, Claudius and Gertrude are alone. They talk about the source of Hamlet's "distemper." Neither mentions the secret of the actual circumstance of King Hamlet's death. If Shakespeare wanted Gertrude to be complicit then this would have been a concern for the two in a moment when only the two are on stage conversing about the cause of Hamlet's strange behavior.
Then of course there is the closet scene where Gertrude appears oblivious to Hamlet's prompting about killing a king.