Morrissey was born in Liverpool to a family of four. He decided to become an actor after seeing the Ken Loach film Kes and appearing in a school production of The Wonderful Wizard of Oz. While a teenager, Morrissey trained at Liverpool's Everyman Youth Theatre, with Paul McGann's brothers Mark and Stephen. At the age of eighteen, he appeared in the Yorkshire Television/Channel 4 drama One Summer before moving to London to train at the Royal Academy of Dramatic Art.