Dennis was born in Seattle, Washington, United States. His mother was a violinist and his father a singer, and the family was in vaudeville, so he was exposed to music early. In 1933 he joined Horace Heidt's orchestra as a vocalist and pianist. Later on, he formed his own band, with Dick Haymes as vocalist. He became vocal coach, arranger, and accompanist for Martha Tilton, and worked with a new vocal group, The Stafford Sisters. Jo Stafford, one of the sisters, joined the Tommy Dorsey band in 1940 and persuaded Dorsey to hire Dennis as arranger and composer. Dennis wrote prolifically, with 14 of his songs recorded by the Dorsey band in one year alone, including "Everything Happens to Me", an early hit for Frank Sinatra..

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