Ulrich was born into an upper-middle-class family in Gentofte, Denmark; the son of Lone (née Sylvester-Hvid) and professional tennis player Torben Ulrich. His paternal grandfather was professional tennis player Einer Ulrich. His paternal grandmother, Ulla Meyer, was from a Jewish family; as a result, Ulrich's grandfather was persecuted by the Nazis during World War II. Saxophonist Dexter Gordon was Ulrich's godfather, and he is a childhood friend of musician Neneh Cherry.
In February 1973, Ulrich's father obtained passes for five of his friends to a Deep Purple concert held in the same Copenhagen stadium as one of his tennis tournaments. When one of the friends could not go, they gave their ticket to the nine-year-old Ulrich, who was mesmerized by the performance and bought the band's album Fireball the next day. The concert and album had a considerable impact on Ulrich, inspiring the start of his music career.
As a result of his newfound interest in music, he received his first drum kit, a Ludwig, from his grandmother around the age of 12 or 13. Ulrich originally intended to follow in his father's footsteps and play tennis, and he moved to Newport Beach, California, in the summer of 1980. Despite being ranked in the top ten tennis players of his age group in Denmark, Ulrich failed to make it into the 7 man Corona del Mar High School tennis team, contributing to his decision to focus on music.
In the documentary Anvil! The story of Anvil, Ulrich states that witnessing a Y&T show was his defining moment in deciding to become a musician. In 1981, he discovered British heavy metal band Diamond Head. He was excited about the band's style of music after purchasing their debut album Lightning to the Nations, and traveled from San Francisco to London to see the band perform live at the Woolwich Odeon. Ulrich remains a fan of Diamond Head and would later go on to mix their album The Best of Diamond Head. Upon returning to America, Ulrich placed an advert in a local classifieds newspaper looking for musicians to start a band with him. James Hetfield replied to the ad, and Metallica was formed.