I too would focus on how Hamlet is upset with his mother because he feels that she has betrayed him, and her kingdom. He is starting to worry about what happened to his father and feel suspicious of his uncle. He lashes out because he feels helpless. I'd note her point of view, which might be that her actions were not so reprehensible because she did not have a choice and was trying to keep the kingdom together in her own way.
With a paper like this, it's a good idea to understand who is doing the talking and why they're saying what they're saying. Hamlet is so angry with his mother for remarrying so quickly after her husband's death...and marrying her brother-in-law to boot! We know Hamlet loves his mother, but he is angry and in despair over her seeming lack of grief for the loss of King Hamlet.
You could compare his reactions to Gertrude at this point in the play to his reactions to her further on, particularly in the scene in her bedchamber where Hamlet kills Polonius (Act 3, scene 4). He goes from one point in the story of just talking to himself (and the audience, of course) about her being so forgetful of her husband (1.2) to openly yelling at her about being with Claudius (3.4). Hamlet definitely goes through some major changes in his willingness to confront her about what he considers to be her sin, and so that might make for a good topic for your paper.
Also, be sure you quote it correctly in your paper - "Frailty, thy name is woman!" (Act 1, scene 2). :)