Constructing $\pi_1$ actions on higher homotopy groups.

One way to think about the action of $\pi_1(X)$ on the higher homotopy groups of $X$ is to think of it as being induced by the action of $\pi_1(X)$ on the universal cover $\widetilde{X}$, which has the same higher homotopy groups as $X$. We can try to reverse this argument, and to construct the desired space by first constructing its universal cover with the desired homotopy groups, then constructing the desired action of $\pi_1(X)$ on it, and finally quotienting by this action appropriately.

Constructing $\widetilde{X}$ is the easiest part: we can just take it to be a product $\prod_{n \ge 2} B^n \pi_n(X)$ of Eilenberg-MacLane spaces (where by $B^n A$ I mean $K(A, n)$). If you believe that the construction of Eilenberg-MacLane spaces is functorial then any desired action of $\pi_1(X)$ on each $\pi_n(X)$ induces an action on $B^n \pi_n(X)$ and hence we get the desired action of $\pi_1(X)$ on $\widetilde{X}$.

The tricky part now to make sure that quotienting $\widetilde{X}$ by $\pi_1(X)$ actually gives a covering map, so that the quotient $X$ actually has the correct homotopy groups. This is what the Borel construction is for; it's a distinguished way to modify $\widetilde{X}$ in a way that preserves both its (weak) homotopy type and the action of $\pi_1(X)$ on it, but so that the action of $\pi_1(X)$ is free.