How does Gertrude highlight the weakness of man and his susceptibility to sins of the flesh?
First, if I were casting for the play, I would cast a rather good-looking actor for the role of Claudius. This would help explain, to some extent, the temptation he posed for Gertrude.
Gertrude seems to take the path of least resistance. Most probably she and Claudius had some kind of relationship before King Hamlet's murder. I base this theory on the fact that Gertrude's marriage to Claudius cinched his position as king, so Claudius must have been pretty certain of Gertrude before killing his brother. Gertrude did grieve over the death of her husband, but she married Claudius very quickly. Her reasons for doing so are not clear: perhaps she was strongly attracted to Claudius, perhaps she was trying to maintain life as she knew it as the queen, perhaps she feared facing her future alone without a man to dominate and control. We rarely know what Gertrude is thinking, so any of these factors could contribute to her relationship with Claudius. But once married to him, she clearly enjoys him. Her openly demonstrative behavior toward him is what angers Hamlet and is central to his diatribe against her in Act 3.
In Act 3, when Hamlet informs her that the one who now shares her bed is the one who murdered her former husband, Gertrude reacts in horror. This moment of truth seems only to paralyze Gertrude, "Thou hast cleft my heart in twain," she tells Hamlet. Gertrude's loyalties are now divided between her son and her husband. Even though her husband is a murderer, how can she break with him? Gertrude is not strong enough to confront the king. Besides, it would mean that her entire world would come crashing down. So, she does nothing. In Act 4, Gertrude stands by the king as if she knows nothing about his heinous crime, even defending him against Laertes' attack. In Act 5, when she is dying from the poisoned cup, she does not speak out against Claudius even while trying to warn Hamlet about the drink.
Gertrude probably represents many of us who try to maintain the status quo when trying to do the right thing would take enormous sacrifice and might even be futile. For Gertrude, speaking out against Claudius, exposing him as a murderer, might mean sacrificing her life and the stability of her country.