Odell was born in Newark, New Jersey, in 1737 to John and Temperance Odell. He graduated from Princeton University (at the time known as the College of New Jersey) in 1754. Although he had studied medicine, instead of becoming a doctor he joined the Church of England ministry. As a minister he preached at parish priest at Burlington and Mount Holly, both in New Jersey.
When the revolution broke out Odell became a strong loyalist and wrote poetry promoting the loyalist cause. He was brought before the New Jersey Provincial Congress for such actions and on July 20, 1776, he was ordered to sign a loyalty oath and remain within eight miles of the Burlington County courthouse. In December of that year, he fled to New York, with the help of local citizens, and served as an administrator and satiric poet-propagandist for the British. After the war in 1784 he emigrated to New Brunswick, Canada, where he received the post of provincial secretary as a reward for his loyalty. He remained in New Brunswick and died in Fredericton.