Welliver studied at the Philadelphia Museum College of Art where he received a B.F.A. in 1953 and at Yale University, earning an M.F.A. in 1955. He was awarded a Morse Fellowship in 1960 and a Skowhegan Award in 1975. Welliver's landscapes have been the subject of a number of one-person exhibitions in New York at Tibor de Nagy Gallery and Fischback Gallery. An exhibition of his work traveled to the Worcester Art Museum, Worcester, Massachusetts; the Des Moines Art Center, Iowa; the Currier Gallery of Art, Manchester, New Hampshire; the Columbus Museum of Art, Ohio; the Institute of Contemporary Art, University of Pennsylvania; and the Virginia Museum of Fine Arts, Richmond.
His paintings are in the collections of the Baltimore Museum of Art; the Museum of Fine Arts, Boston; the Metropolitan Museum of Art, New York; the Museum of Modern Art, New York; and the Frances Lehman Loeb Art Center, Vassar College, Poughkeepsie, New York. Welliver was on the faculty at Cooper Union, New York from 1953-1957, Yale University, New Haven, from 1955-1965, and the University of Pennsylvania, Philadelphia from 1966-1978. Welliver's large-scale images of the Maine woods, which typically portray the cold light of winter or early spring, are inspired by the 1,200 acres of wilderness on which he lives near Lincolnville, Maine.