Hickey was born in Plano, Texas, and graduated from Plano Sr. High School in 1981. He attended Texas State University – San Marcos from 1981–1983, where he was active in the theater department. He earned his bachelor’s degree in English at Fordham University in 1985.

On Broadway, he originated the role of Arthur in Terrence McNally’s play Love! Valour! Compassion! in 1995, a role he recreated for the 1997 film version.

He played Clifford Bradshaw in the 1998 Broadway revival of Cabaret, which won the Tony for Best Revival of a Musical. In that same year, he played the lead in the independent film Finding North.

On Broadway, he played Reverend John Hale in the 2002 revival of Arthur Miller’s The Crucible.

Hickey played American novelist and playwright Jack Dunphy in the 2006 Truman Capote biopic Infamous. Hickey played Philip Stoddard on the short-lived gay-themed sitcom It’s All Relative. Since It’s All Relative, Hickey has appeared on Alias, Law & Order, Brothers & Sisters, Stacked, Heartland, In Plain Sight, Law & Order: Los Angeles, and Modern Family.

From 2010 to 2013, he appeared on The Big C and was nominated for a Primetime Emmy Award for Outstanding Supporting Actor in a Miniseries or a Movie for the final season of the series, subtitled Hereafter. Hickey starred as scientist Frank Winter on the TV series Manhattan, which concluded on December 15, 2015 after two seasons and 23 episodes.

He appeared in the 2011 Broadway revival of The Normal Heart, for which he won the Tony Award, Featured Actor in a Play. He was in the Broadway revival of Mary Stuart in 2009, as the Earl of Leicester.

In 2015, he appeared Off-Broadway at the Mitzi Newhouse Theatre in the play Dada Woof Papa Hot by Peter Parnell.

In 2018 he appeared in the world premiere of Matthew Lopez’s new play The Inheritance, inspired by the novel Howards End by E.M. Forster, creating the role Henry Wilcox at London’s Young Vic and then transferring to the West End.