 Arbitrary-degree maps to a surface of genus two

For $k>0$, an epimorphism ${\pi }_1({\Sigma }_2)\to \mathbb{Z}/k$ gives a connected $k$-sheeted cover. Its Euler characteristic is $-2k$, hence its genus is $k+1$, and the lifted orientation gives degree $k$. For negative $k$ compose the $|k|$-sheeted cover with an orientation-reversing homeomorphism of the target. For $k=0$ take a constant map from the torus.