Two definitions of a degree of map $f: S^n \to S^n$ equivalent?

There is a Hurewicz homomorphism $\Phi:\pi_n(S^n) \to H_n(S^n)$ given by taking a class of a map $\pi_n(S^n)\ni[\alpha]:S^n \to S^n$ and sending it to $\alpha_*([S^n])$ where $[S^n]$ is the fundamental class, or the generator for $H_n(S^n)$. It is an isomorphism here.

This map is natural in the sense that for any map $f:S^n \to S^n$ and $[\alpha] \in \pi_n(X)$ we have that $\Phi(f_!([\alpha])=f_!(\Phi([\alpha]))$.

The point here is that $\Phi$ preserves degree! This buys you that definition two is equivalent to the induced homomorphism $f_*:H_n(S^n) \to H_n(S^n)$

To make this leap to the first definition, you will need to make the use of the De-Rham isomorphism.