He was educated at The Westminster Schools and attended the University of Georgia on a baseball scholarship. After attending a recital by Jessye Norman and being cast as Judge Turpin in a college production of Sweeney Todd, he decided to leave university after his sophomore year in order to study voice at the Manhattan School of Music where he majored in opera and graduated in 1989. From there he went to the Curtis Institute of Music, Philadelphia, and obtained his master's degree in 1993.
Hensley's stage career began in the early 1990s with roles such as Pitkin in On the Town, Joe in The Most Happy Fella, and Miles Gloriosus in A Funny Thing Happened on the Way to the Forum, He has also sung in the Gilbert and Sullivan operettas Pirates of Penzance and Patience and in the operas Carmen, Faust, La bohème and Don Giovanni. And in addition to an early Oklahoma! casting as Jud Fry at the North Shore Music Theatre, Boston, Shuler also played Curly at the Skylight Opera Theatre, Milwaukee.
He has performed on several occasions with Hugh Jackman. They both performed in the musical Oklahoma! in London in 1998, as well as in the films Someone Like You and Van Helsing, in the latter of which Hensley played Frankenstein's Monster.
In 2011 he portrayed American Yiddish theatre great Boris Thomashefsky in The Thomashefskys: Music and Memories of a Life in the Yiddish Theater, a concert stage show celebrating the Thomashefskys and the music of Yiddish theatre, hosted by their grandson the Conductor Michael Tilson Thomas. The show aired on the PBS series Great Performances in 2012.
In 2012 he starred as a 600-pound man in the Off Broadway production of The Whale at Playwrights Horizons.
The Georgia High School Musical Theatre Awards, part of the National High School Musical Theatre Awards – The Jimmy Nederlander Awards, are named in his honor.
In fall 2016, Hensley co-starred with Sutton Foster in The New Group's revival of Sweet Charity.
Hensley also starred in the Broadway production of Young Frankenstein as the Monster. For this role he was nominated for the Drama Desk Award for Outstanding Featured Actor in a Musical and the Outer Critics Circle Award for Outstanding Featured Actor in a Musical. He also reprised the role in the United States national tour which also featured his former co-stars Roger Bart and Cory English. He again played the role in the 2017 West End production.
