Murray is a graduate of the University of Virginia, where she was a Jefferson Scholar and an Echols Scholar, and Yale Law School, where she was notes development editor of the Yale Law Journal. While in law school, she earned special recognition as an NAACP-LDF/Shearman & Sterling Scholar and was a semifinalist of Morris Tyler Moot Court.

Following law school, Murray clerked for Sonia Sotomayor, then of the US Court of Appeals for the Second Circuit, and Stefan Underhill of the US District Court for the District of Connecticut. Murray is a member of the New York bar.

Murray teaches constitutional law, family law, criminal law, and reproductive rights and justice, among other courses. Murray’s research focuses on the legal regulation of sex and sexuality and encompasses such topics as marriage and its alternatives, the marriage equality debate, the legal recognition of caregiving, and reproductive rights and justice. Her publications have appeared (or are forthcoming) in the California Law Review, Columbia Law Review, Harvard Law Review, Michigan Law Review, Pennsylvania Law Review, Virginia Law Review, and Yale Law Journal, among others. She is an author of Cases on Reproductive Rights and Justice, the first casebook to cover the field of reproductive rights and justice. She has translated her scholarly writing for more popular audiences by publishing in the New York Times, Newsweek, the San Francisco Chronicle, Vanity Fair, and the Huffington Post, and has offered commentary for numerous media outlets, including NPR, MSNBC, and PBS.

In 2013, Murray's article, "What's So New About the New Illegitimacy?," was awarded the Dukeminier Awards' Michael Cunningham Prize as one of the best sexual orientation and gender identity law review articles of 2012. Her article, "Marriage as Punishment," won the Association of American Law Schools' 2010-2011 Scholarly Papers Competition for faculty members with fewer than five years of law teaching. "Marriage as Punishment" was also selected by the Association of American Law Schools' Section on Women in Legal Education as a winner of the 2010-2011 New Voices in Gender Studies scholarly paper competition. In 2010, Murray was awarded the Association of American Law School's Derrick A. Bell Award, which is given to a junior faculty member who has made an extraordinary contribution to legal education, the legal system, or social justice. In 2011, Murray was elected to the membership of the American Law Institute.

Prior to joining the NYU faculty, Murray was on the faculty of the University of California, Berkeley, School of Law, where she was the recipient of the Rutter Award for Teaching Distinction. From March 2016 to June 2017, she served as interim dean of the Berkeley Law.