Is the Converse of the Cayley-Hamilton Theorem true or not? I was thinking about taking the Cayley-Hamilton Theorem in the opposite direction.
Does this hold : If for $F\in$ End$(V)$ $\exists P\in K[t]$ such that $P(F)=0$ then 
$P$ is the characteristic polynomial of $F$ ?
If not could you give a counterexample?

No. If $p_F$ is the characteristic polynomial of $F$ and $q$ is any other polynomial, then
$$P(F) = p_F(F) q(F) = 0 \cdot q(F) = 0.$$
So $P$ is a polynomial with $P(F) = 0$ that is not the characteristic polynomial of $F$ (just take $q$ to be nontrivial).
We can also have something like the following happen: Let
$$F = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}.$$
Then
$$p_F(t) = (1 - t)^2.$$
But of course if
$$q(t) = 1 - t,$$
we have
$$q(F) = 0.$$
