Alfred Clebsch was born in 1833 to Ernst Friedrich Leopold Clebsch and Pauline Ramberg in Konigsberg, Prussia (now Germany). He attended the University of Konigsberg, where he was introduced to mathematical physics by his professor and friend's father, Franz Neumann. In 1854 he received his doctorate for his thesis that explored a problem in hydrodynamics. After four years of teaching at a secondary school, he took up his first positions in higher education at Humboldt University of Berlin and then the University of Karlsruhe.

He married Dorothe Charlote Mathilde Heinel (1838-1866) while teaching at Karlsruhe, and they later had four sons. Before leaving Karlsruhe in 1863, the direction of Clebsch's work began going more towards mathematics than physics, particularly the calculus of variations. He collaborated with Paul Gordan until 1868 while he was a professor at the University of Giessen. During his time with Gordan, he worked in Abelian function, modern algebraic geometry, and invariant theory. The two also developed the Clebsch-Gordon coefficients, which are used in spherical harmonics. In 1868 he and his friend Carl Neumann, son of Franz Neumann, founded the "Mathematische Annalen," a mathematical research journal.

In his later years Clebsch worked more on binary invariants until his premature death due to diphtheria in 1872 at age 39.