Genesis records nothing more about Asenath, but her story is elaborated in the apocryphal Joseph and Asenath. There, she is a virgin who rejects several worthy suitors in favor of Joseph, but Joseph will not have a pagan for a wife. She locks herself in a tower and rejects her idolatry in favor of Joseph's God Yahweh, and receives a visit from an angel who accepts her conversion. A ritual involving a honeycomb follows, and Joseph now consents to marry her. She bears him their sons Mannaseh and Ephraim. Pharaoh's son wants Asenath for himself, however, and with the aid of Joseph's brothers Dan and Gad, he conspires to kill her husband. The loyal brother Benjamin interferes, and Pharaoh's son is ultimately slain. Asenath forgives the conspirators, and she and Joseph rule over Egypt for 48 years, after which they pass the crown to Pharaoh's grandson.
The Midrash Pirke De-Rabbi Eliezer records a view that Asenath was actually the daughter of Joseph's sister Dinah, conceived in her rape by Shechem. Some of the circumstantial textual evidence supporting this view is laid out in an article by John Pratt.
"Asenath" or "Osnat" is a commonly used female first name in present-day Israel.