Kellum gained his first major role in 2012 when he was cast in the FOX comedy Ben and Kate (then titled Ben Fox Is My Manny) playing the role of Tommy, the best friend of the titular character Ben (Nat Faxon).[7] Following Ben and Kate, Kellum was cast as Hunter in NBC's Sean Saves the World starring Sean Hayes and Megan Hilty.

In 2015, Kellum was cast in the recurring role of Curtis Holt, based on the DC Comics character Mister Terrific, in the fourth season of The CW drama series Arrow.[8][9] Holt is a technological savant and inventor who initially works at Palmer Technologies under the supervision of Felicity Smoak, and later becomes a member of Team Arrow. Kellum also went on to portray the character in the Arrow spin-off series Legends of Tomorrow. In April 2016, it was announced that Kellum would be promoted to series regular for the fifth season of Arrow.[10] Kellum exited the series during its seventh season in 2019.[11] He returned for two episodes in Arrow's final season.[12]

In 2017,[13] Kellum appeared in Netflix's Girlfriend's Day[14] as Madsen, beat poet and a co-worker of Bob Odenkirk's character.[15]

Kellum voices the character "King Joaquín" on the Disney Channel animated series Elena of Avalor. He also voices multiple characters for the Adult Swim animated series Rick and Morty.

In 2021, Kellum landed a role on the NBC series Grand Crew.