Gertrude's weakness would be that she uses poor judgement. She so desires to keep her position as queen until she marries Claudius to maintain her position. She did not give herself time to mourn the death of her husband King Hamlet. She did not consider her son's feelings in the matter.
Less than two months after her husband's death, Gertrude marries Claudius. Hamlet, her son, is ouraged. He comments that his mother has quickly rushed into an incestuous relationship (I.ii 156-157).Truly, Gertrude does have a character flaw in that she does not analyze the consequences of her marriage to Claudius. She does not consider her son's sensitivity at the time. Hamlet believed she should be mourning his father's death instead of marrying his brother. She uses poor judgment by rushing into the marriage.