Kosteck studied at the University of Maryland at College Park and obtained his Bachelor of Music in 1959 in music theory and composition. He then studied with Leslie Bassett and Ross Lee Finney at the University of Michigan in Ann Arbor and received his Master of Music in music theory, composition / music history and piano in 1961. With a scholarship from the Fulbright program, he was able to study in 1962 and 1963. the Conservatory of Amsterdam with Ton de Leeuw and with Jan Odé. He once again completed his studies at the University of Michigan in Ann Arbor and graduated there in 1964 as Doctor of Musical Arts in composition with his work Rhapsody for cello and orchestra, in musicology and piano.
He became a lecturer at the Washington and Jefferson College in Washington in 1965/1966. From 1966 to 1973 he was a lecturer at East Carolina University in Greenville. There two of his operas also premiered. He then spent two years teaching at the Appalachian State University in Boone. Then followed two years (1974-1976) at Columbia University in New York City. He taught at the University of Tennessee in Knoxville in 1976 and 1977.
Then he was a freelance composer and teacher in New York City. He received several national and international prizes and awards for his compositions such as the prize at the Festival dei Due Mondi in Spoleto (Italy) (1965), in 1969 the composition prize at the International Queen Elisabeth Competition in Belgium, in 1972 the Concours Quatuor au Cordes Prix International in Liège, in 1976 the Premio Internacional de Composición Óscar Esplá in Alicante (Spain), in 1977 the Harvey Gaul Opera Prize, in 1978 a prize at the International Music Festival for Orchestral Music in Costa Rica, in 1980 the New York State Arts Council Award, in 1981 a prize at the International Henryk Wieniawski Composers' Competition in Poznań  and in 1982 a special prize from the American Society of Composers, Authors and Publishers (ASCAP). As a result of his illness (sids) he could hardly compose anymore from the middle of the 1980s.
He was married to Nancy Elizabeth Bender.