He attended Frankfort High School where he began to pursue his interest in the theatre arts, and wrote poetry and prose for the school's literary journal. After high school, Wolfe enrolled at the historically-black Kentucky State University, the alma mater of his parents, Costello and Anna Lindsey Wolfe. Following his first year, he transferred to Pomona College in Claremont, California, where he pursued a BA in theater. Wolfe then taught for several years in Los Angeles at the Inner-City Cultural Center and later in New York, where he received an MFA in dramatic writing and musical theater at New York University in 1983.
Wolfe gained a national reputation with his 1991 musical Jelly's Last Jam, a musical about the life of jazz musician Jelly Roll Morton; after a Los Angeles opening, the play moved to Broadway, where it received 11 Tony nominations. Two years later, Wolfe directed Tony Kushner's Angels in America: Millennium Approaches to great critical acclaim as well as a Tony award. Wolfe also directed the world premiere of the second part of "Angels", entitled Perestroika, the following year.
From 1993 to 2004, Wolfe served as artistic director and producer of the New York Shakespeare Festival/Public Theater, where, in 1996, he created the musical Bring in 'Da Noise, Bring in 'Da Funk, an ensemble of tap and music starring Savion Glover; the show moved to Broadway's Ambassador Theatre and his work won him a second Tony Award for direction and was an enormous financial success.
In 2000, Wolfe co-wrote the book and directed the Broadway production The Wild Party.
In late 2004, Wolfe announced his intention to leave the theater for film direction, beginning with the well-received HBO film Lackawanna Blues.
Despite this move, Wolfe continues to direct plays such as Tony Kushner's Caroline, or Change and Suzan-Lori Parks' Pulitzer Prize-winning play Topdog/Underdog. In the summer of 2006 he directed a new translation of Bertolt Brecht's Mother Courage and Her Children at the Delacorte Theatre in Central Park starring Meryl Streep, Kevin Kline, and Austin Pendleton.
Wolfe is openly gay.