Wait...Gertrude and Claudius weren't seeing each other while King Hamlet was still alive were they?I read something saying that the reason Claudius...

Hamlet does call Claudius an "adulterous beast," but I don't think this implies that Claudius and Gertrude were having an affair while the king was still alive. The Ghost blames Claudius completely, saying he seduced his queen but the Ghost does of course hold a grudge against his queen. The Ghost's language makes it seem like there was something going on while he was still alive, but this is because the Ghost is still a part of this world, in a kind of purgatory. Since the Ghost is still around, bearing witness to Claudius and Gertrude's relationship, it is as if, from his (Ghost's) perspective, he is still alive. So, the Ghost's language might infer that he 'is being' betrayed, when in fact, he was not betrayed by Gertrude until after his death (albeit too soon after his death).

Also, the Ghost tells Hamlet not to harm Gertrude: vengeance is marked only for Claudius. Gertrude's penance is that she must deal with her conscience. Most evidence points to Gertrude being shallow and a bit naive, but at times she is portrayed as inherently good in wishing for the well being of those around her: particularly Hamlet and Ophelia, but also Claudius. She takes too long to see what's going on, but there are too many hints that say she was not involved in the plot.

In ACT III, Scene iii, Claudius says:

"Of those effects for which I did the murder,

My crown, mine own ambition and my queen.

May one be pardon'd and retain the offence?"

This could be the order of his plot, the seduction of the queen being last, implying the queen was not involved and they were not having an affair until after Claudius had killed the king and achieved the crown.