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.