
Not a great deal is known about Astell's life. According to Ruth Perry's bio, The Celebrated Mary Astell: An Early English Feminist (University of Chicago Press, 1986), "as a woman she had little or no business in the world of commerce, politics or law. She was born; she died; she owned a small house for some years; she kept a bank account; she helped to open a charity school in Chelsea." Only four of her letters still exist, yet she is still considered an important early advocate of women's education and independence.
She was born at Newcastle-upon-Tyne in 1666 to Peter and Mary Astell, who had two sons as well, one of which died in infancy and the other her younger brother Peter Junior. They were comfortably upper-middle-class (if there can be said to be such a thing at the time!), her father a conservative Royalist and devout Anglican who managed a coal company. Mary received most of her early education from her uncle, an ex-clergyman who wa suspended from the Church for drunkeness. Her father died when she was 12, leaving the family in financial straits, and Mary and her mother moved in with her aunt.

Astell died May 11, 1731, after a mastectomy to remove a cancerous right breast. She refused in her final days to see anyone, staying alone in her dark room with her coffin to commune with God (or something like that).
A Serious Proposal to the Ladies is available in an edition from the Broadview Press. A few other sources are:
Christine Sutherland, The Eloquence of Mary Astell (2005)
Ed. William Kolbrener and Michael Michelson, Mary Astell: Reason, Gender, Faith (2007)