In Hartshorne's, he states that #$C\cap D=deg_C(\mathcal{F}(D)\otimes\mathcal{O}_C)$, where $deg_C$ denotes the degree of the invertible sheaf $\mathcal{F}(D)\otimes\mathcal{O}_C$.
I couldn't find any explicit definition about degree of sheaf in the book, but there are some implications. Since divisor $D$ and the corresponding invertible sheaf $\mathcal{F}(D)$ is one to one. So the degree of an invertible sheaf means the degree of the corresponding divisor. Is that right?
Yes that's correct; alternatively you can define the degree of any coherent sheaf $\mathcal F$ as
$$\deg\mathcal F=\chi(C,\mathcal F)-(\operatorname{rank}\mathcal F)\,\chi(C,\mathcal O_C)$$
and notice that this is the same as what you're describing in the case that $\mathcal F$ is an invertible sheaf by using Riemann-Roch.