Calculating Euler Characteristic of Closed Subscheme

Question

Suppose $X$ is a projective k-scheme of dimension at least one. I want to know how to compute $\chi(X,\mathscr{O}_X(-d))$, the Euler characteristic. My idea was to use the short exact sequence $0 \to J_X \to \mathscr{O} \to \mathscr{O}_X \to 0$ where $\mathscr{O}$ is the structure sheaf on projective space and $J_X$ is the sheaf of ideals of X in $\mathbb{P}^n_k$. But now I don't really know anything about the sheaf of ideals, so using the additivity of short exact sequences doesn't get me anywhere. Can anyone give me a hint on this problem?