What's the relation between Voevodsky's motivic cohomology and Quillen Ktheory of a scheme?
You should look at Marc Levine's preprint "Ktheory and motivic cohomology of schemes, I". The version on the UIUC Ktheory server seems to be older than the version on his website .
Roughly speaking, the motivic spectral sequence starts from motivic cohomology and converges to algebraic Ktheory. This spectral sequence was conjectured by Beilinson and first written down for nice fields by Bloch and Lichtenbaum (preprint on the UIUC server), and was extended to more general schemes by Friedlander and Suslin (Ann. Sci. École Norm. Sup. (4) 35 (2002), no. 6, 773875). I think these papers were written before the bulk of Voevodsky's work, and the E_2 term was described in terms of Bloch's higher Chow groups. The correspondence between motivic cohomology and higher Chow groups is Theorem 1.2 in Levine's paper, and is due to Voevodsky ("Motivic cohomology groups are isomorphic to higher Chow groups in any characteristic," Int. Math. Res. Not. 2002, no. 7, 351355).

1The spectral sequence exists for all schemes (say, over a field) if one uses higher Chow groups and K'theory. It then gives the desired result for smooth schemes because higher Chow groups are isomorphic to motivic cohomology and K'theory to Ktheory. Also, Marc Levine has several papers where he gives a proof of the spectral sequence without referring to the BlochLichtenbaum paper which never appeared. – Thomas Geisser Sep 25 '12 at 8:19
I found that for a smooth quasiprojective variety $k$: $CH^q(X,p)_\mathbf{Q} = K_p(X)^{(q)}_\mathbf{Q}$, where $CH^q(X,p) = H^{2qp,q}(X)$ are Bloch's higher Chow groups.