Kapičić was born December 1, 1978 in Cologne, North Rhine-Westphalia, West Germany, the son of Slobodanka "Beba" Zugić, a Montenegrin Actress, and Dragan Kapičić, a retired Serbian basketball player. His father played for Saturn Köln during that time. At three years of age the family returned to Serbia, SFR Yugoslavia. His grandfather bought him his first comic book when he was six years old, which sparked what he called an "ongoing love" in a 2015 interview about his role in Deadpool. He studied acting at the Faculty of Dramatic Arts in Belgrade. He speaks Serbian, Croatian, English, German and Russian.

In December 2015, Kapičić became engaged to Croatian Actress Ivana Horvat. They got married in 2017.

By 2015, Kapičić had accumulated more than 80 roles in television, film, and theater. He made it to the second round of a secret casting process for the X-Man Colossus in the superhero film Deadpool, learning what the production was only after he passed that round. Working with first-time Director Tim Miller in Los Angeles, Kapičić did 120 takes during his third call of auditions, stretching the session an additional eight hours beyond its intended four-hour duration, before flying back to Dubrovnik, Croatia to perform Romeo and Juliet. Kapičić, a self-described "comic book geek" who considered Deadpool one of his favorite characters, was especially delighted when he learned he won the role, taking over from Daniel Cudmore, who had played the role in the X-Men films. Kapicic was selected because Miller wanted his version of Colossus, who is Russian in the comics, to have a Russian accent. The 6-foot-4 Kapičić portrayed the 7-and-a-half-foot tall character through a combination of voice work and motion capture, completing filming in mid-December 2015, eight weeks before the film's February 12, 2016 release date.