The homotopy category of chain complexes over a ring has the same objects as the category of chain complexes and has chain homomorphisms modulo chain homotopy as its morphisms. Composition of morphisms is induced by composition of morphisms of chain homomorphisms. Formally making its quasi-isomorphisms invertible gives the derived category.
More generally, a differential graded category 
has a homotopy category 
whose hom-sets are the degree-zero
cohomology groups of its morphism
complexes. For the differential graded
category of complexes, closed degree-zero morphisms
are chain homomorphisms and degree-zero boundaries
are chain-homotopic to zero, recovering the preceding
definition (Stacks Project, Definition 22.26.3 and Example 22.26.6).
