Polonius advises Ophelia that men use "springes to catch woodcocks," referring to the tricks men use to seduce women when he warns Ophelia to stay away from Hamlet. This trap imagery that is prevalent throughout the play is particularly relevant in terms of the female characters. Both Ophelia and Gertrude are trapped in a patriarchal society. Many of their decisions can be read in this light. The two female characters are linked by Gertrude being the one to report Ophelia's death, as well as by Hamlet's disdain of first his mother and then Ophelia. "Frailty, thy name is woman," Hamlet cries out. But when you look at the social structure that Shakespeare establishes in the play, it is easy to see that the women are frail because so much is beyond their control.
I think that this topic would find a great deal of resonance in Shakespeare's play. It is evident that "the role of women" topic is something that has been discussed or distilled in your class and you should probably integrate the full extent of these ideas throughout your paper, not merely in the introduction. I do think that you would probably find much in way of discussion in Gertrude and Ophelia. The latter would be an excellent starting point to discuss how Shakespeare displays women being manipulated from others. Ophelia is manipulated by her father and Hamlet, and to a lesser extent, her brother. She is an individual who presents her voice to those around her, but is silenced through men's actions. The fact that she is the only character who displays her emotions with a sense of authenticity and essentially receives rebuke for it is something to explore in terms of how women are treated both in the play and in social settings, in general. At the same time, Gertrude receives scorn from Hamlet for making herself happy after the death of her husband. The question that might be worthy of exploration here would be if she would have received the same treatment had it been a man who remarried after his wife had died. These valences of gender can help to make a very persuasive analysis about the role of women as shown in the play.