Born in Braunschweig, Koch received violin lessons from Ion Voicu in Berlin.
In 1955 he became director of the master class at the Hochschule für Musik Freiburg, from 1967 professor for viola himself.
Well-known Koch students include Hatto Beyerle, Ruth Killius [de], Vincent Royer, Henrik Schaefer, Hermann Voss and Tabea Zimmermann.
Recordings with works by Paul Hindemith, Karl Amadeus Hartmann, Béla Bartók, Jean Françaix, Darius Milhaud, Raymond Baervoets, Karel Husa, Johann Nepomuk David, Miklós Rózsa document his manifold work.
Since the nineties, an "Ulrich Koch Prize" has been awarded to outstanding young viola players within the Wolfgang Marschner International Competition.