Is his mother really remorseful or does she only pretend to be because of her fear of Hamlet?Act 3 sc 4
I believe Gertrude is truly remorseful. Hamlet accuses her of marrying too quickly...too conveniently...and in a way, who can blame her? Women in Shakespeare's time were only considered "valid" members of society based upon the men to whom they were attached. Even noble-born women needed a husband, brother, or son to take care of them and the business of living. The marriage to Claudius after King Hamlet's death served two purposes--Gertrude remains Queen and ruler of Denmark, thus continues to live richly as she is accustomed; Claudius gets his ultimate goal of the title of KING. It is quite possible that Gertrude didn't think King Hamlet's death through, as well. His death appeared to all as one of natural causes. It is possible that she never entertained the idea that he was actually murdered, and murdered by the man to whom she is now married, until young Hamlet comes to her bed chamber and poses the question.
With all this in mind, I believe Gertrude is indeed remorseful.
His mother is afraid that Hamlet has gone mad, but I think she also begins to understand what she has done and why Hamlet is so upset. So it's a little of both. But, Gertrude doesn't tell Claudius after Hamlet asks her not to, so I think this shows she is remorseful for her actions.