The degree of vector bundle on integral projective curve is the degree of its determinant bundle. This is exercise 18.4.J in Vakil's "The Rising Sea  Foundations of Algebraic Geometry".
Here the degree of a coherent sheaf $\mathscr{F} $ on integral projective curve $ C $ is defined to be
$$ \mathrm{deg} \mathscr{F}=\chi(C,\mathscr{F})-(\mathrm{rank} \mathscr{F})\cdot \chi(C,\mathscr{O}_C).$$
Here the curve may be singular, and thus the local ring at a closed point is not a discrete valuation ring. I have no idea whether we can define the order of zeros or poles of a rational section in this case.
A hint is given: Exercise 13.5.H, i.e., given an exact sequence
$$0\to \mathscr{F}_1\to \cdots \to \mathscr{F}_n \to 0 $$
of finite rank locally free sheaves on $ X $, the alternating product of determinant bundle is trivial.
My idea is that since $\mathscr{F}(m) $ is gloally generated for $m>>0 $, we get an exact sequence of locally free sheaf
$$ 0\to \mathscr{F_0} \to \oplus\mathscr{O}(-m)\to \mathscr{F}\to 0,$$
but I don't know whether $\mathscr{F_0}$ is direct sum of line bundles, and I'm not sure if we still have
$$\mathrm{deg}(\mathscr{F}\otimes\mathscr{G})=\mathrm{deg} \mathscr{F}+\mathrm{deg} \mathscr{G}$$
in this case. When C is regular, this is done by computing the divisor of zeros and poles of a rational section, but here $ C $ may be singular now.
 A: Let $\mathcal{F}$ be a rank $n$ vector bundle (i.e. locally free sheaf) on such a curve $C$. Note that $\det(\mathcal{F})$ is a locally free sheaf of rank 1 (an invertible sheaf). We want to show that $\deg \mathcal{F} = \deg (\det \mathcal{F} ) $, i.e.
$$ \chi(C, \mathcal{F}) - n \cdot \chi(C, \mathcal{O}_C) = \chi(C, \det(\mathcal{F})) - \chi(C, \mathcal{O}_C), $$
or equivalently,
$$ \chi(C, \mathcal{F}) = \chi(C, \det(\mathcal{F})) + (n-1) \chi(C, \mathcal{O}_C). $$
Recall that Euler characteristic is additive in exact sequences, so we would be done if we can find some exact sequence
$$ 0 \to ? \to \mathcal{F} \to \det ( \mathcal{F} ) \to 0 $$
or
$$ 0 \to \det ( \mathcal{F} ) \to \mathcal{F} \to ? \to 0 $$
where $\chi(C, ?) = (n-1) \chi(C, \mathcal{O}_C)$. Thinking about the definition of $\det{\mathcal{F}}$, can you find such a sequence? The hint given in the text might give you some guidance but Vakil's hints can also be distracting at times.
Another route: use a result like https://stacks.math.columbia.edu/tag/0AYT, bullet point (4).
