Tveit is known for originating the roles of Gabe in Next to Normal and Frank Abagnale Jr. in the stage version of Catch Me If You Can. He is also known for his recurring role in The CW's teen drama Gossip Girl as Tripp, Nate Archibald's cousin and Serena van der Woodsen's ex-lover, and for his portrayal of revolutionary student Enjolras in the 2012 film adaptation of Les Misérables (2012 Film).
Aaron was born in New York. He graduated from Middletown High School in 2001, where he was active both in theatre and sports, playing golf, soccer and basketball while also performing in all four of his school's theatre productions. He turned down business school scholarships to major in vocal performance at Ithaca College, a decision his parents supported.
Aaron left Ithaca College after two years to join the national tour of Rent as Steve and covering Roger/Mark. Following Rent, he was cast as Link Larkin in the first national tour of Hairspray.
The role of Link Larkin provided Tveit with his Broadway debut in 2006. In the following year, he portrayed D'Artagnan in a musical adaptation of The Three Musketeers that played at the North Shore Music Theatre from late August to early September. In July 2007 Tveit played Matt in the Barrington Stage Company's Calvin Berger.
In 2008 he starred as Gabe in the Off-Broadway production of the musical Next to Normal, which ran from January 16 through March 16 at the Second Stage Theatre. Aaron received a nomination for the Lucille Lortel Award for Outstanding Featured Actor for this role. Also in June 2008, Aaron began performances as Fiyero in Wicked. He left the role in November in order to reprise his role in the Arena Stage production of Next to Normal, for which he was awarded the 2009 Helen Hayes Award for Outstanding Supporting Performer in a Non-Resident Production.
Aaron returned to Wicked on January 2009, leaving once again for the Broadway production of Next to Normal, which began previews in March 2009 and officially opened on April 15. Aaron's performance as Gabe earned him the Clarence Derwent Award from the Actors' Equity Association. Aaron left Next to Normal on June 6, 2009 to prepare for Catch Me If You Can, where he played Frank Abagnale Jr.. Aaron returned to Next to Normal on September 7 through January 3, 2010. Tveit landed on AfterElton.com's list of the "37 Hottest Guys in Theater". He also starred in the Hollywood Bowl's production of Rent directed by Neil Patrick Harris.
Aaron also made appearances on the CW series Gossip Girl as Trip Van Der Bilt, the cousin of Nate Archibald. He appeared in the television series Ugly Betty for one episode, entitled "All the World's a Stage". He portrayed Zachary Boule, Betty's boyfriend.
Aaron played Frank Abagnale Jr in the Broadway production of Catch Me If You Can, opening on April 10, 2011 and closing on September 4, 2011. For this role he was nominated for the Outer Critics Circle Award for Outstanding Actor in a Musical, the Drama League Award for Distinguished Performance, and the Fred Astaire Award for Best Male Dancer on Broadway.
He was also featured in the April 2011 Vanity Fair magazine for his role in Catch Me if You Can.