Camp was born in Aiken, South Carolina. Her mother, Dee (née Kornegay), is a Democratic Party volunteer, and her father, Thomas Sewell Camp, is a bank executive. Her elder sister, Saluda, is also an actress.
She grew up in Columbia, South Carolina. She attended Meadowfield Elementary School and was cast in a Drug Abuse Resistance Education production in the second grade, introducing her to acting. She graduated from the University of North Carolina School of the Arts with a Bachelor of Fine Arts degree in 2004. She moved to New York City shortly thereafter.
Camp played the role of Perfect in columbinus in 2005. In 2007, she was nominated for a Lucille Lortel Award for her performance in the Off-Broadway play The Scene. She played Jill Mason in the 2008 Broadway revival of Equus at the Broadhurst Theatre, which starred Daniel Radcliffe as Alan Strang. Speaking about the role, Camp said, "I had a lot of thought; I didn't even know if I was going to do Equus because of the nudity and because of the high profile [aspect] of it. But you only live once and you have to take those risks because you'll only be a better person or actor because of it."
She appeared in Reinventing the Wheelers, a 2007 television pilot which was not ordered to series by ABC. In 2008, she had a role in the pilot of the ABC television dramedy Cashmere Mafia. She had a starring role as Sarah Newlin in the second season of the HBO supernatural drama series True Blood, which earned her a nomination for a SAG Award for Outstanding Performance by an Ensemble in a Drama Series; Camp had earlier auditioned for the role of Sookie Stackhouse.
Camp has made guest appearances on The Office (2009), Glee (2009), Numb3rs (2010), and Covert Affairs (2010). She has had recurring roles in the fourth season of the AMC drama series Mad Men (2010) and the third season of the CBS legal drama series The Good Wife (2011–2012) and How I Met Your Mother (2013).
In 2011, Camp starred in the premiere of All New People, a play written by Zach Braff and staged at Second Stage Theatre; the production was directed by Peter DuBois and also starred Justin Bartha, David Wilson Barnes and Krysten Ritter.
Camp had a starring role in the first season of Fox sitcom The Mindy Project, and played Aubrey Posen in the 2012 musical comedy film Pitch Perfect and reprised her role in Pitch Perfect 2.
Most recently, she has a recurring role in Netflix's Unbreakable Kimmy Schmidt as Deirdre Robespierre.
Camp was engaged to actor Michael Mosley by September 2008; they married in 2010 and filed for divorce in 2013. Camp began dating Pitch Perfect co-star Skylar Astin in 2013. The couple were reported to be engaged in January 2016. They married on September 10, 2016. Camp appeared in an episode of Astin's series, Ground Floor, as his character's ex-girlfriend."Pitch Perfect [Soundtrack Album]" Released: September 25, 2012
"Starships" ft. Pitch Perfect Cast and Mike Tompkins Released: October 2012