As $\mathcal{O}_{\mathbb{P}^2}$ is trivial, then multiplicativity of Chern classes in exact sequences implies:
$$
c_*(\mathcal{E}) = c_*(\mathcal{I}_p(-1)).
$$
We can compute $c_*(\mathcal{I}_p(-1))$ by noting that $p\in \mathbb{P}^2$ is a complete intersection of 2 lines. Therefore, the $(-1)$-twist of the Koszul complex is exact:
$$
0\rightarrow \mathcal{O}_{\mathbb{P}^2} (-3) \rightarrow \mathcal{O}_{\mathbb{P}^2}(-2)^{\oplus 2} \rightarrow \mathcal{I}_p(-1)\rightarrow 0.
$$
Thus
$$
c_*(I_p(-1)) = c_*(\mathcal{O}_{\mathbb{P}^2}(-2)^{\oplus 2})/c_*(\mathcal{O}_{\mathbb{P}^2}(-3)) = (1-4h+4h^2)/(1-3h)\\
=(1-4h+4h^2)(1+3h+9h^2)\\
= 1-h+(9-12+4)h^2 = 1-h+h^2,
$$
where $h\in H^2(\mathbb{P}^2,\mathbb{Z})$ is the hyperplane generator. Thus: $c_1(\mathcal{E})=-1$ and $c_2(\mathcal{E}) = 1$.

I.e. $\int_{[\mathbb{P}^2]} c_2(\mathcal{E}) = 1$.

One can also use Grothendieck-Riemann-Roch to compute $c_*(\mathcal{O}_p) = 1-h^2$, and then use the ideal sequence for $\mathcal{O}_p$.