$H_2(X)$ is all about $\pi_1(X)$ and $\pi_2(X)$. If $\pi_2(X)$ is trivial (as for knot complements) then it is a functor of $\pi_1(X)$.
Let $H_n(G)$ be $H_n(BG)$, the homology of the classifying space ($K(G,1)$). If $X$ is path-connected than there is a surjection $H_2(X)\to H_2(\pi_1(X))$ whose kernel is a quotient of $\pi_2(X)$, the cokernel of a map from $H_3(\pi_1(X))$ to the largest quotient of $\pi_2(X)$ on which the canonical action of $\pi_1(X)$ becomes trivial.
This $H_2(G)$ isn't anything like the next piece of the derived series after $H_1(G)=G^{ab}$, though. For example, if $G$ is abelian then $H_2(G)$ is the second exterior power of $H_1(G)$ (EDIT: so it can be nontrivial even though it knows no more than $H_1(G)$ does), while if $H_1(G)$ is trivial $H_2(G)$ is often nontrivial (EDIT: so, even when it does carry some more information than $H_1(G)$, it is not necessarily derived-series information).
EDIT: The previous paragraph comes from looking at the integral homology Serre spectral sequence of $X\to K(\pi_1(X),1)$, where the homotopy fiber is the universal cover $\tilde X$. Since $H_1\tilde X=0$, the groups $E^\infty_{p,1}$ are trivial and we get an exact sequence
$$
0\to E^\infty_{0,2}\to H_2(X)\to E^\infty_{2,0}\to 0,
$$
therefore
$$
E^2_{3,0} \to E^2_{0,2}\to H_2(X)\to E^2_{2,0}\to 0.
$$
Since $H_2(\tilde X)=\pi_2(\tilde X)=\pi_2(X)$, this looks like
$$
H_3(\pi_1(X)) \to H_0(\pi_1(X);\pi_2(X))\to H_2(X)\to H_2(\pi_1(X))\to 0.
$$
The place to look for the rest of the derived series would be homology with nontrivial coefficients, for example homology of covering spaces.
The following construction is due to Jones and Westbury, in a paper titled "Algebraic $K$-Theory, homology spheres and the $\eta$-invariant" (it is a very nice paper).
Let $M$ be a homology $3$-sphere and $\rho:\pi_1 (M) \to GL_N(\mathbb{C})$ a representation. Then the Quillen plus construction gives a map $S^3 = M^+ \to (BGL_N(\mathbb{C}))^+$, in other words, an element $[M,\rho] \in K_3 (\mathbb{C})$. On this group, there is the $e$-invariant $e:K_3 (\mathbb{C})\to \mathbb{C}/\mathbb{Z}$. Jones and Westbury give a formula for $e([M,\rho])$ in terms of the eigenvalues of $\rho$. If $M$ is the Poincare sphere, you get a complicated, but manageable formula.
Now replace $\rho$ by an action of $\pi_1 (M)$ on a finite set $X$. This then gives, by Barratt–Priddy–Quillen, an element of $\pi_{3}^{st}$. The Jones–Westbury formula tells you how to compute the e-invariant of the image of this element in $K_3 (\mathbb{C})$ under the map induced by $\Sigma_N \to GL_N(\mathbb{C})$. As the homomorphism $\pi_{3}^{st} \to K_3 (\mathbb{C})$ is injective, you do not lose information.
As a grad student, I played with these formulae in the similar situation of mapping class groups (instead of general linear or symmetric group). This gives elements in the homotopy of the plus-construction of the classifying space of the mapping class group and I was able to find a generator of $\pi_3 (B\Gamma^{+})=\mathbb{Z}/24$ in this way (by an acion of the tgroup of the Poincare sphere on a surface of rather small genus, see http://wwwmath.uni-muenster.de/mjm/vol3.html. I guess it is possible to get a generator of $\pi_{3}^{st}=\mathbb{Z}/24$ by an action of the fundamental group of the Poincaré sphere on a finite set.
Best Answer
Try looking up some references on rational homotopy theory. Rational homotopy theory studies the homotopy groups tensor Q, so basically you kill all torsion information. If we focus only on homotopy groups tensor Q, the question you ask becomes easier. As Steven Sam mentions in the comments, the homotopy groups of spheres are really crazy. But the rational homotopy groups of spheres are quite tractable (in fact completely known, by a theorem of Serre) and can be more or less obtained from cohomology, if I recall correctly.
One particularly impressive theorem, of Deligne-Griffiths-Morgan-Sullivan, says that if your space is a compact Kähler manifold (e.g. a smooth complex projective variety), and if you know its rational cohomology ring, then you can compute for instance the ranks of all of its homotopy groups (maybe you need an extra assumption that the space is simply connected or has nilpotent fundamental group).