How would I analyze the character of the Queen? What kind of a person is she?
Assuming that Queen Gertrude did not know Claudius killed her husband, either before or after the fact, then her precipitate marriage to Claudius might not have been as outrageous as her son Hamlet thinks it was. If Claudius proposed to her, she had to say either yes or no. If she declined, then her position would have been precarious. No doubt she would have been permitted to remain in the palace, but she would have been treated like a poor relation, stripped of servants, ladies in waiting, and all the other amenities she was accustomed to. Furthermore, her son would no longer have been the heir apparent to the throne. Hamlet would be another poor relation, treated almost like a pauper or even as a servant. Claudius would have wanted to marry someone. He needed a queen, if only to strengthen his image as the ruler. He would probably have tried to marry the daughter of some other monarch or foreign nobleman, as was so customary in those days. His new wife would have probably been quite young. They would probably have had several children, and all of the children would have had precedence to the crown over Hamlet. No doubt Gertrude thought about all these factors when Claudius proposed to her. Gertrude may have been more concerned about the future of her son than about herself. Another possibility was that Claudius might actually marry the young, attractive Ophelia. If he suggested it to her father, Polonius would be delighted and had the power to force his submissive daughter to accept.
There's not all that much to reveal. She's simple, naive, superficial, non-introspective, and easily influenced. She's a bauble, an ornament for a King. She's emotional and rather innocent. Indeed, the two women in the play, Gertrude and Ophelia, are very much alike, and what Hamlet says to Ophelia about women could apply to both of them (Act 3, scene 1):
I have heard of your paintings too, well enough. God hath given you one face, and you make yourselves another. You jig, you amble, and you lisp; and nickname God's creatures and make your wantonness your ignorance.
Certainly not a very flattering point of view, but very close to the truth as it pertains to the two women in Hamlet's life.