Hawkshaw was born in Dublin, and entered Trinity College, Dublin in 1687. He left Ireland upon the revolution, and entered St. John's College, Cambridge, graduating B.A. there in 1691. He subsequently returned to Dublin, where he proceeded B.A. in 1693 and M.A. two years afterwards. He took orders, and was appointed to the parish of St. Nicholas-within-the-Walls at Dublin, a rectory in the gift of the Corporation of Dublin. Hawkshaw held four rectories in North Tipperary, in the Diocese of Killaloe (Nenagh, Monsea, Kneagh (Knigh), Killodiernan from 1720–38, and may also have held two rectories in diocese of Kilmacud. He died in 1738.