This is not a complete answer to your question, which I am not sure how to answer without having a precise meaning for what you mean by "computing" and what you mean by "knowing" $\pi_1(X)$, as indicated in the comments above, and given that you have not included the data of any action of $\pi_1(X)$.

However, I think you may be interested in taking a look at my paper with Mahmoud Zeinalian "Singular chains and the fundamental group".

In this article, we prove that for any path connected pointed space $(X,b)$, the $E_{\infty}$-coalgebra structure (in fact, the $E_{2}$ part of it) of (a pointed version of) the singular chains $C_*(X,b)$ extending the Alexander-Whitney coproduct determines $\pi_1(X,b)$ functorially and in complete generality. The correct notion of weak equivalence under which this information is preserved is defined via the cobar functor.

In particular, the natural algebraic structure on the singular chains $C_*(X,b)$ determines a chain complex (by purely algebraic means) which calculates the homology of the universal cover of $X$ at $b$. The construction is the following: consider the differential graded connected coassociative coalgebra $C=(C_*(X,b), \partial, \Delta)$ and take its cobar construction $\Omega C$. This is a dg associative algebra such that $H_0(\Omega C)\cong \mathbb{Z}[\pi_1(X,b)]$, the fundamental group ring. There is a natural "twisting cochain" (in the sense of Brown) $\tau: C \to H_0(\Omega C)$ through which we may construct a twisted tensor product $(C \otimes_{\tau} H_0(\Omega C), \partial_{\tau})$. The homology of this chain complex is the homology of the universal cover. Note that the differential $\partial_{\tau}$ uses both the dg coassociative coalgebra structure of $C$ and the algebra structure of $H_0(\Omega C)$ (which in turn, was obtained from the dg coassociative coalgebra structure of $C$). Also, this construction is invariant in the following "Koszul duality" sense: if $f: C\to C'$ is a map of dg coalgebras such that $\Omega f: \Omega C \to \Omega C'$ is a quasi-isomorphism of dg algebras then the induced map $$(C \otimes_{\tau} H_0(\Omega C), \partial_{\tau}) \to (C' \otimes_{\tau} H_0(\Omega C'), \partial_{\tau})$$ is a quasi-isomorphism. This notion is strictly stronger that ordinary quasi-isomorphisms of dg coalgebras. In some sense, this suggests that you need more information than just the homology to determine the action of $\pi_1$ on the universal cover.

This fact can be used to show an extension of classical theorem of Whitehead, namely we can now prove that a continuous map of pointed path connected spaces $f: (X,b) \to (Y,c)$ is a weak homotopy equivalence if and only if the induced map at the level of dg coassociative coalgebras of pointed singular chains $f: C_*(X,b) \to C_*(Y,c)$ becomes a quasi-isomorphism after applying the cobar functor.

Note that I didn't use the homotopy groups of $X$ directly but I used more information than the homology of $X$, namely, the chain level natural algebraic structure of the singular chains. Again, the question of "how computable this is" is a different one, which I am not sure how to formulate precisely (and it may be tricky given that group theory is not decidable).