I would like to upper bound the probability $$Pr[X \leq C \cdot E[X]]$$ where the upper bound involves only the expectation and the constant $C$. The best I could find is in the case $X \leq B$, ($B$ may be another r.v. or a constant) for all $X$, where we have
$$ Pr[ X \leq C \cdot E[X] ] \leq \frac{E[B-X]}{B-C\cdot E[X]} $$
(Cor 2 in http://www.cs.ubc.ca/~nickhar/W12/Lecture2Notes.pdf )
The random variable X is discrete and takes a finite number of values (I want to avoid an upper bound that depends on the size of the domain).
There may be no upper bound for $\Pr(X \leq C \cdot E[X])$ apart from $1$.
Consider $\Pr(X=0)=1-\frac1n$ and $\Pr(X=n)=\frac{\mu}{n}$ and then let $n$ increase
As for a lower bound when $X$ is non-negative, try Markov's inequality to get $\Pr(X \leq C \cdot E[X]) \ge 1 - \frac1C$