I'll try to sketch a proof of Riemann-Hurwitz using the Leray spectral sequence. It has the feel of a fun exercise.

To fix notation, let $X$ and $Y$ be compact Riemann surfaces and let $f : X \to Y$ be a finite surjective morphism of degree $d$. We want to compare the topological Euler numbers of these surfaces; these are the numbers defined as
$$
\chi(X) = h^0(X,\mathbb C) - h^1(X,\mathbb C) + h^2(X,\mathbb C)
$$
and similarly for $Y$. We'll use arbitrarily fancy facts of sheaf cohomology to do this.

If $\mathcal F$ is a sheaf on $X$, then the first terms of the Leray spectral sequence read
$$
E_2^{p,q} = H^q(Y, \mathcal R^p f_* \mathcal F) \Rightarrow H^{p+q}(X,\mathcal F).
$$
As the fibers of $f$ are 0-dimensional, we have $\mathcal R^p f_* \mathcal F = 0$ for any $p \geq 1$. Combined with the annihilation of cohomology on $Y$ for dimension reasons, we find that $E^{p,q}_2 = 0$ for any $p \geq 1$ and $q \geq 3$. The second page of the Leray spectral sequence is thus just
$$
E_2^{0,0} \qquad E_2^{0,1} \qquad E_2^{0,2}
$$
and all other entries are zero, so the sequence degenerates at the $E_2$-level. It follows that $H^k(Y,f_*\mathcal F) = H^k(X,\mathcal F)$ for any $k$.

Consider now a point $y$ on $Y$ that is not in the image of the ramification locus of $f$, in other words the preimage $f^{-1}(y)$ consists of $d$ distinct points. Then we see that $f_{\ast} {\mathbb C} = {\mathbb C}^{\oplus d}$. This line of though yields a short exact sequence
$$
0 \longrightarrow f_* \mathbb C \longrightarrow \mathbb C^{\oplus d} \longrightarrow
\mathcal G \longrightarrow 0
$$
where $\mathcal G$ is a skyskraper sheaf supported on the image of the ramification divisor of the morphism $f$. Taking Euler characteristics we get
$$
d\,\chi(Y) = \chi(f_*\mathbb C) + \chi(\mathcal G)
= \chi(X) + h^0(Y,\mathcal G).$$
Expressing $h^0(Y,\mathcal G)$ in terms of the degrees of $f$ at its ramification points, and thus showing that it has the expected form, should not be a source of great trouble.

The thing that makes this proof relatively painless is that the Leray spectral sequence degenerates straight away (at least without recourse to heavy machinery) and that calculating the cohomology of a sheaf supported on a finite number of points is easy. The spectral sequence will again degenerate at the $E_2$-level in the case of a morphism between surfaces, but there a finer analysis is needed to calculate the cohomology of the corresponding sheaf $\mathcal G$. In any case the proof points the way to a similar statement for finite surjective morphisms between higher dimensional varieties, though it also seems to indicate that this is not a path one wants to take unless one really needs to.