Law was born at Crosshouse in Ayrshire. Law went to school in Kilmarnock and took evening classes at the Kilmarnock Academy. In 1891, he was awarded the National Medal for Success in Art and won a place at the Glasgow School of Art. In 1896, Law was awarded a travelling scholarship and spent six months studying in Paris, where he took lessons from Robert Henri and attended classes at the Academie Delecluse. Law returned to Kilmarnock and began a career as a professional portrait painter. He also taught part-time at the Glasgow School of Art and exhibited regularly at the Royal Scottish Academy.