  Let $\mathcal{D}$ be a weighted oriented graph and $I(\mathcal{D})$ be its
edge ideal. In this paper, we investigate the Betti numbers of $I(\mathcal{D})$
via upper-Koszul simplicial complexes, Betti splittings and the mapping cone
construction. In particular, we provide recursive formulas for the Betti
numbers of edge ideals of several classes of weighted oriented graphs. We also
identify classes of weighted oriented graphs whose edge ideals have a unique
extremal Betti number which allows us to compute the regularity and projective
dimension for the identified classes. Furthermore, we characterize the
structure of a weighted oriented graph $\mathcal{D}$ on $n$ vertices such that
$\textrm{pdim } (R/I(\mathcal{D}))=n$ where $R=k[x_1,\ldots, x_n]$.
