In Fulton's book "Intersection Theory", Theorem 15.3 and Example 15.3.1 (p. 297) it is proven, by using GRR, that $$c_k(\mathscr{O}_Z)=(-1)^{k-1}(k-1)![Z].$$
By the short exact sequence
$$0 \to \mathscr{I}_Z \to \mathscr{O}_X \to \mathscr{O}_Z \to 0$$
one has $1=c(\mathscr{O}_X)=c(\mathscr{O}_Z)c(\mathscr{I}_Z)$ and this implies $$c_k(\mathscr{I}_Z)=(-1)^k (k-1)![Z].$$