Elizabeth I and other female "leaders of state" during her time were very different, and in order to compare Gertrude to these women, it would be best to study predecessors and/or contemporaries of Elizabeth—they would have been familiar figures to Shakespeare.
Men had been ruling England for a long time. Prior to the war between Britain and Normandy (France), the area was ruled by the Anglo-Saxons. When the Normans invaded, the Saxons were defeated, and this began a new era where two different cultures were joined. William the Conqueror was the first "French" king to sit on the throne in Great Britain—in 1066. Perhaps one of the strongest women to hold influence over England was Eleanor of Aquitaine. First married to Prince Louis VII of France, they divorced because Eleanor had not produced a male heir. Later she married Henry II, Duke of the Normans—even while other noblemen were trying to kidnap her and take her lands and wealth.
Eleanor bore her husband eight children (including five sons). Henry was unfaithful and Eleanor grew unhappy with the marriage. She was politically savvy—later offering support to her son Henry III when he tried to take the throne from his father. The revolt was a failure and Eleanor was imprisoned for sixteen years. Henry flaunted his mistress in Eleanor's face, hoping she would agree to an annulment; in doing so, she would forfeit her lands and wealth to her husband. Eleanor was much too smart for this. She outlived Henry and all but two of her children.
Many years later, when Henry VIII died, his nine-year old son Edward VI took the throne, guided by his Regency Council. On his deathbed, he named Lady Jane Grey, his cousin, as heir. Though intelligent, her actions were controlled by husband John Dudley and others. She reigned for nine days, but was imprisoned. Treasonous acts committed by Thomas Wyatt the Younger led to the execution of Jane and her husband.
Forces supporting Mary I, first daughter of Henry VIII, brought Mary to the throne. She was a staunch Catholic. She married Philip II of Spain (to form an alliance), and persecuted the Protestants. (She was known as "Bloody Mary.") She had her sister Elizabeth imprisoned for many years. Mary I was strong in her own right, but she needed the military support of her husband to keep her power. when Mary I died only five years later, Elizabeth I was released and warmly welcomed into England. Where Eleanor had ruled on behalf of her son who left England to fight in the Crusades, Elizabeth I was a Queen in her own right, and never married. She was greatly admired and brought peace (eventually) and prosperity to England. She was a softer version of her father, but could rule like a man when necessary.
Gertrude, on the other hand, seems to be nothing like these other women...except for Lady Jane Grey—their similarity is that each seemed governed by her husband. Where Eleanor marries a second time and gains power in her own right (having title, lands and wealth), Gertrude marries Claudius but gains nothing that we know of, and has no power. She tries her best through her personal relationship with the King to protect her son. Where Mary was brutal, Gertrude was not: as seen in how she cares for Hamlet and Ophelia. Unfortunately, Gertrude is treated simply like a woman and dependent upon a man; Elizabeth depended on her strength of character and quick thinking.
Gertrude, like Lady Jane Grey, was more a pawn and martyr than a strong woman and queen.