If $X$ is a Markov process with continuous paths, then the additional condition for it to be a diffusion and hence a Feller process is that it be strong Markov (see the book by Ito-McKean, Section 3.1). For example, your example is not strong Markov (stop it when it hits 0).
If you only require $X$ to be a Markov process with not necessarily continuous paths, then your condition is (almost) equivalent to requiring that 1) $S_t:C_0(E) \rightarrow C_0(E)$, where $(S_t)_{t\ge0}$ is the semigroup of $X$ and $C_0(E)$ is the space of continuous functions vanishing at infinity. But a Feller process requires moreover that 2) $S_tf(x) \rightarrow f(x)$ for each $x$, as $t\rightarrow 0$ (see for example Section III.3 in the book by Revuz-Yor). For example, take Brownian motion on $\mathbb R^+$ which jumps to $1$ when it hits $0$. This process satisfies 1) but not 2)
If you take for $X$ any stochastic process, I doubt that one can tell more than what you wrote in your comment.