If $\mathcal F^\bullet$ is a chain complex of sheaves, to compute the hypercohomology, you take the cohomology of an injective resolution of $\mathcal F^\bullet$, i.e., a chain complex of injectives $\mathcal I^\bullet$ which is quasi-isomorphic to $\mathcal F^\bullet$.

It seems to be a well-known fact that for the algebraic de Rham complex $\Omega^\bullet_{X/S}$ you can compute this hypercohomolgy in practice by taking the total cohomology of the Cech/de Rham double complex $(\check{C}^i(X,\Omega^j), d, \check{d})$.

Does anyone know of a reference for this (somewhere in EGA?), or is there an easy proof ?