Bremner was born in Edinburgh, the son of two art teachers. He attended Davidson's Mains Primary School and Portobello High School. He originally wanted to be a circus clown, but was offered a chance in show business by television director Richard D. Brooks. One of his first notable roles was as a Glasgow schoolboy in Charles Gormley's Heavenly Pursuits (1986). He would also play the lead in the BBC Scotland feature-length adaptation of the William McIlvanney short story "Dreaming" (1990).