We can interpret the slope of vector bundle over a curve of genus $g$ using Riemann-Roch formula. Let me explain it
From of differential geometric point of view the degree of a holomorphic vector bundle can be computed by Chern–Weil formula in terms of curvature, and the fact that curvature decreases in sub-bundles. We explain Chern-Weil formula which gives an effective way for degree of holomorphic vector bundle.
A reflexive sheaf (i.e its double dual is equal itself) is locally free (i.e., a holomorphic vector bundle) outside a subvariety of codimension greater than or equal to two. Let $\mathcal F$ be a coherent subsheaf of holomorphic vector bundle $E$, then there is an analytic subset $S \subset M$ of codimension bigger than two and a holomorphic vector bundle $F$ on $X \setminus S$ such that $$\mathcal F|_{X\setminus S}=\mathcal O(F)$$ and $F$ is a sub-bundle of $E|_{X\setminus S}$ and there is an orthogonal projection $\pi:E|_{X\setminus S}\to F$ which $\pi\in L_1^2(End(E))$ lying in the Sobolev space of $L^2$ sections of $End(E)$ with $L^2$ first-order weak derivatives and satisfying $\pi=\pi^*=\pi^2$ where $\pi^*$ denotes the adjoint of $\pi$. The Chern-Weil formula is $$deg_\omega \mathcal F=\frac{\sqrt[]{-1}}{2\pi n}\int_X tr(\pi\Lambda_\omega F_h)\omega^n-\frac{1}{2\pi n}\int_X|\nabla''\pi|^2\omega^n$$ where $\nabla''\pi$ is computed in the sense of currents using the $(0,1)$ part of the Chern connection of $E$.
We define the slope of $\mathcal E$, to be $$\mu(\mathcal E)=\frac{deg \mathcal E}{rk \mathcal E}$$
For any non-trivial vector bundle $E$ on curve $X$ by using the Riemann-Roch formula we can compute the slope of a vector bundle over a curve as follows,
$$\mu(E)=\frac{dim H^0(X,E)-dim H^1(X,E)}{rank E}+g_X-1$$ where $g_X$ is the genus of curve $X$.