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

$$   c_k(N) = ( \nu_k -\nu_{k-1} ) h^k. $$

As for the  integers $\nu_k$  they are determined from the equality


$$ 1+c_1( \mathbb{CP}^{n-1} )+\cdots + c_{n-1}( \mathbb{CP}^{n-1} )= (1+H)^n - H^n $$

$$= \sum_{k=0}^{n-1}\binom{n}{k} H^k. $$