We generalize the techniques developed in the previous paper [10] on free algebras and free bimodules to path algebras and projective bimodules. We develop the theory of Gr\"{o}ber bases on path algebras and their projective bimodules, and use it to construct projective resolutions of bimodules over a quotient algebra of a path algebra. It gives an effective way to calculate the Hochschild cohomology of algebras expressed as quotients of path algebras. We also give a formula for the cup product in the cohomology in terms of our resolution. It gives a way to determine the ring structure of the cohomology.