Leray's theory of spectral sequences considers a continuous map $f : X \to Y$ between topological spaces. The statement is that there exists a filtration $$H^n(X,k) = F^0H^n(X,k) \supseteq ... \supseteq F^nH^n(X,k),$$ $k$ being a ring, such that the subquotients of this filtration equals the terms $E_{\infty}^{p,n-p}$ of a spectral sequence which has $E_2$-term $$E_2^{p,q} = H^p(Y,R^qf_* k),$$ where $R^qf_*k$ denotes the sheafification of the presheaf $U \to H^q(f^{-1}(U),k)$.

Where can I find a genuine construction of this filtration ? Is there a way to express it in terms of the skeletons of $Y$ (in the CW-complex case), as for the Serre spectral sequence ? I would like to check that a certain homomorphism of cochain complexes induces a map between Leray spectral sequences, and thus I need to check that the map in question preserves this filtration.

Thanks in advance.