In some poems, it's tough to tell who your speaker is and what kind of a person he or she is (or even if it's a man or a woman). Not here. We learn a lot about this woman. Actually, let's start there. This speaker is a woman, and that matters a lot. She talks for most of the poem about what it means to be a woman in her day and age, how it limits her speech, and allows people to make unfair assumptions about her: "I am obnoxious to each carping tongue/ Who says my hand a needle better fits" (25-26). As far as she is concerned, her critics can't even begin to look past the fact that she's woman, or imagine that a woman could do something other than sew.
That's not the only (or even the most important) thing about her, though. Our speaker is also a writer, and proud to be one. She's a passionate reader, well-educated (knows her Greek myths), witty and humble (sometimes to a fault). She's ready to do battle with the people who attack her, but also gracious enough to let things go once she's made her point. She skewers her woman-hating opponents, by reminding them that the Greeks thought goddesses were the inspiration for all poetry: "But sure the antique Greeks were far more mild,/ Else of our Sex, why feigned they those nine" (31-32). Still, even though she has a great point, she's not going to be a jerk about it, and eventually she drops the whole fight: "Let Greeks be Greeks, and Women what they are" (37). That's the speaker of this poem in a nutshell: intelligent and persuasive, but also gracious and thoughtful. All in all, we'd say that she's a pretty cool lady.