The case of unbounded (cochain) complexes and colimits is treated with great detail in the paper:
Alonso Tarrío, Leovigildo; Jeremías López, Ana; Souto Salorio, María José Localization in categories of complexes and unbounded resolutions. Canad. J. Math. 52 (2000), no. 2, 225–247.
(Sorry for the self-reference). The theory is developed in detail in section 2 of this paper but it is not checked explicitly that they correspond to the usual homotopy colimits by the Dold-Kan correspondence.