Second Chern class of a ruled surface

Let $S$ be a ruled surface over a curve of genus $g$. Is it possible to compute the second Chern class of $S$ in terms of $g$?

Sure why not. Riemann-Roch for surfaces gives $1+p_a=\frac{1}{12}(K^2+c_2)$, see Appendix A, Example 4.1.2 in Hartshorne. You also have $K^2=8(1-g)$ and $p_a=-g$, see V.2.11 and V.2.4 respectively in Hartshorne. Putting this together you get that $c_2=4(1-g)$.

Here's another approach.

As $c_2$ is the top Chern class, it is equal to the Euler class which, when paired with the fundamental class, evaluates to the Euler characteristic. Every ruled surface $S$ is a $\mathbb{CP}^1$-bundle over $\Sigma_g$ for some $g$, and therefore has Euler characteristic

$$\chi(S) = \chi(\mathbb{CP}^1)\chi(\Sigma_g) = 2(2 - 2g) = 4(1-g).$$

So $c_2(S)$ is $4(1-g)$ times the oriented generator of $H^4(S; \mathbb{Z})$.