Assume $X$ is smooth compact of dimension $n$ and $x_0\in X$ is the point where we perfrom the blowup. Set $ X_* := X \setminus x_0 $, $ \tilde{X}_* := \tilde{X} \setminus E$. Denote by $N$ a tubular neighborhood of $E$ in $\tilde{X}_* $. By Mayer-Vietoris, the Chern classes of $ \tilde{X} $ are determined once we know their restrictions to $ X_* $ and $ N $.
We identify $ \tilde{X}_* $ with $ X_* $ via the blowdown map $p:\tilde{X}_* \to X_* $. The restriction of $c_k( \tilde{X}) $ to $X_*$ is equal to the restriction of $c_k(X)$. The restriction of $c_k(X_*)$ to $N$ is easy to determine since
$$TN \cong \pi^* T\mathbb{CP}^{n-1} \oplus \pi^* H^*, $$
where $\pi: N\to E= \mathbb{CP}^{n-1}$ is the natural projection and $H\to \mathbb{CP}^{n-1}$ is the hyperplane line bundle. Thus,
$$ c_k(\tilde{X})|_N = c_k( N ) = \pi^*c_k(\mathbb{CP}^{n-1} ) +\pi^* c_{k-1}(\mathbb{CP}^{n-1} ) \pi^* c_1(H^*) $$
$$ = \pi^*c_k(\mathbb{CP}^{n-1} ) - \pi^* c_{k-1}(\mathbb{CP}^{n-1} )\cup \pi^*[H]. $$
Things can be simplified a bit if we introduce the notation $h=\pi^*[H]\in H^2(N,\mathbb{Z})$ and we observe that for some integers $\nu_k$ and $\nu_{k-1}$
$$ \pi^*c_k(\mathbb{CP}^{n-1} ) =\nu_k h^k, $$
$$ \pi^* c_{k-1}(\mathbb{CP}^{n-1} )=\nu_{k-1} h^{k-1}. $$
Then
$$ c_k(N) = ( \nu_k -\nu_{k-1} ) h^k. $$