How to apply the central Limit theorem? I am interested on upper bounding the following probability as $n$ goes to infinity.
\begin{equation}
\mathbb{P} \left\lbrace  \Big|(  \xi_{n}- \mathbb{E} \xi_{n})\Big|> \ell \right\rbrace
\end{equation}
 where,  $\mathbb{E}$ is the expectation, the term  $\ell$ is a positive  number and  $\xi_{n}$ is a random variable that satisfy the Central Limit Theorem:
Suppose $r_{n} \to  0$ but  $\mathrm{lim \  inf}  (nr_{n}^d) > 0$, then
\begin{equation*}
\underset{x \in \mathbb{R}}{\sup} \Bigg| \mathbb{P} \left[ \dfrac{\xi_{n}-\mathbb{E}\xi_{n}}{\sqrt{Var (\xi_{n})}} \leq x \right] - \Phi(x) \Bigg| =  \mathcal{O}(r_{n}) \ \ \ \ \ \ as  \ \ \ \ n \longrightarrow \infty
\end{equation*}
In particular, $(\xi_{n}-\mathbb{E}\xi_{n} )/ \sqrt{Var (\xi_{n})}  \overset{\mathcal{D}}{\longrightarrow} \mathcal{N}(0,1)$, where $\mathcal{N}(0,1)$ denotes a random variable with distribution function $\Phi$. 
The central limit theorem is given for large $n$ so I wonder how to apply it when $n$ is fixed?
 A: That is a nice question and the easy answer is, just apply it. But then you might ask, what is a big enough $n$ and when can we apply it?
That's a difficult question and there are some guidelines out there that people use in practice. For example $n > 30$ is, I believe, considered big enough in most cases. However, in theory, you can always find example such that a simple choice of $n$ does not work.
In your case you could do one of the two things, in my opinion:
(1) Use some hypothesis test, for example KS-test, to test the hypothesis whether or not the random variable minus the mean scaled by the variance is normally distributed.
(2) Don't use the central limit theorem. If you want to bound the probability, you could use Markov/Chebyshev inequality. The only quantity that you then need to calculate is the expectation (or variance) of $|\xi_n - \mathbb{E}(\xi_n)|$. However, this bound will usually be more rough than using the Gaussian approximation the central limit theorem provides.
I hope this helps! If you have any questions, please let me know!
