There's Sapho and she was one of the best lyric poets that the Greeks ever produced. The Greek's absolutely loved her as did the Romans, If I recall correctly Ovid, Horace and Catullus was a huge fan and Plato described her as the "tenth Muse". That said, she is quite difficult to read cause she wrote in a very obscure Aeolian dialect which even people at the time found quite odd. She also uses a very uncommon meters like Sapphic meters for which she is almost unique). While she surely produced large amounts of poetry most of it has gotten lost over time. It certainly did not help that the early church were not at all keen on her and went out of their way to burn her poetry when they could. All we have now are a series of fragments (often from Greek papyrus preserved in Egypt) and quotations from her in other works which have survived. However, what we do have really does prove that she was exceptional, once you get over the fact that the words don't conjugate she writes with a real breadth, clarity and depth of emotion which it is hard to find elsewhere in classical poetry.

Her poetry are almost completely love poems, she is after all a lyric poet, and are sexually very ambiguous. She writes in highly passionate terms about people of both sexes but most often women who came to her seeking instruction. She would then write them highly passionate poems and when they left she would compose the hymns for their weddings.

Agrippina the Younger, mother of Nero, wrote and published an autobiography that was used as a primary source for some later historians. But, sadly, it has been lost.

There are surviving letters and pieces of poems from Roman women. And we know of Roman female poets whose work has been completely lost, such as Cornificia. Rome and Greece were both highly patriarchal societies, although women had more relative freedom and education in Rome than generally in the Greek world, so women were not really encouraged to write much. But the Greek Sappho was one of the greatest poets to ever live. The Greeks counted her as just below Homer.

Anna Komnene is outside your time period (she wrote around 1150) but she was the daughter of a "Byzantine"/Roman emperor. And she wrote a history of her times, which includes the First Crusade and the Great Schism.

If you're willing to count late antiquity as part of the classical world, then the rise of christianity actually provided some opportunity for women (usually already rich though) to express themselves and be circulated amongst the community.

The two main female autobiographies that come to mind are Perpetua and Egeria. Perpetua was a rich christian noblewoman who kept a diary while in prison before being sent into the amphitheatre to die around 200 CE, so before the Christianization of the empire.

Egeria (no one is entirely sure of her name actually) was a traveler/pilgrim who wrote the details of her journey to the holy land in what's presumed the late 4th century. It's a fascinating account of the eastern empire by someone who wasn't intending for her diary to be a specifically religious text, so it's been used quite often as a sort of proto-ethnography of the early Christian empire.