Shakespeare uses imagery and diction to reveal Antony and Cleopatra. If you look at the words and images that are used to describe Antony, or man, they are typically structured and civilized words that connote order. The words and images that are used to describe Cleopatra, or woman, are wild, exotic, and based in the natural world. The contrast shows the two sexes as equally powerful, but with power that comes from different places. Man's power comes from the structure and order of society. Woman's power comes from the beauty and abundance of nature.
You could write a doctoral thesis on Shakespeare’s attitude towards women there is so much contrasting material to work with. The one thing that is beyond doubt is that he never described them as equals, even though he created such strong characters as Helena (All’s Well That Ends Well) and Beatrice (Much Ado about Nothing).
The contrast between Rome and Egypt in Anthony and Cleopatra is very much one of masculine versus feminine. Octavius is the proper Roman, strong, self-disciplined, humorless and intelligent. Cleopatra is the Eastern temptress, sensual, clinging, self-indulgent and weak. She causes the downfall of the once proper Roman, Anthony, because she cannot help herself. It is her nature, just it was Eve’s nature to tempt Adam and cause his downfall.
How much Shakespeare really knew about either Anthony or Cleopatra is a guess. He had access to the standard Roman historians, who would have told him that Cleopatra reigned against all the odds for 16 years from the age of 18. Shakespeare could have made her the female equivalent of Octavian. Probably that would have shocked and displeased the audience of the day, and Shakespeare knew his audience.
Whether any of this reflects Shakespeare’s own view of women, we will never know. This is the man who wrote both Hero and Beatrice into Much Ado About Nothing. It is his incredible ability to write into his plays whatever *you* want to see that makes him so amazing.