Born in Leipzig, Roch attended the Polytechnic Institute (Technische Bildungsanstalt) in Dresden, initially focusing on chemistry, encouraged by his father.

Later in 1859, Roch entered the University of Leipzig, coming under the influence of August Ferdinand Möbius, and continuing his work on electromagnetism.

In 1861, Roch went to work at the University of Göttingen, studying under Wilhelm Eduard Weber, but also attending lectures by Bernhard Riemann.

After three terms in Göttingen, Roch went to the University of Berlin, where he met Leopold Kronecker, Karl Weierstrass and others.

In the following year he published the paper containing the result for which he is famous to this day, the Riemann–Roch theorem (given its name by Max Noether), which related the topological genus of a Riemann surface to purely algebraic properties.