This is a very tough question. I don't think that you are likely to find useful necessary and sufficient conditions for the existence of an invariant volume form, especially if you require your volume forms to be smooth instead of just, say, absolutely continuous. There is one pretty general theorem that comes to mind that seems related to your topological criterion.
Let $f\colon M\to M$ be a $C^k$-smooth map, with $k\geq 2$. Suppose that there exists a Riemannian metric on $M$ with respect to which $f$ is expanding. That is, there exists a $\chi>1$ such that $\|Df(v)\|\geq \chi\|v\|$ for all tangent vectors $v$ on $M$. Then $f$ has a unique absolutely continuous invariant measure $\mu$, and this measure is strictly positive. Moreover, $\mu$ is at least $C^{k-1}$-smooth. In particular, if $f$ is $C^\infty$, the measure $\mu$ is a smooth volume form.
I believe the theorem remains true when $k = 1$ as well, except that $\mu$ may no longer be strictly positive. I could be wrong about that though. The expanding condition definitely implies your topological condition, so in that way this is a weaker statement than what you are hoping for. Then again, I don't think the statement you are hoping for is actually true. For a reference to the above theorem, see the article The measures invariant under an expanding map by Richard Sacksteder.
A word about your idea for finding an invariant volume form: it can be pushed a little further. Suppose $\omega$ is the volume form you start with, normalized to give you a probability measure $\mu$. Let $\mu_n$ be the measure $\mu_n = \frac{1}{n}\sum_{k=0}^{n-1} f^{k}_*\mu.$ This is a probability measure for each $n\geq 1$, but, as you say, there is no guarantee that the $\mu_n$ converge to anything, much less a volume form. However, because the weak topology on the space of Radon probability measures on $M$ is compact, there is some subsequence $\mu_{n_k}$ that converges weakly to a probability measure $\nu$. This $\nu$ is easily seen to be invariant. Of course, it will usually not be a volume form.
One final comment as far as conditions on the existence of an invariant volume form go: one thing that must be true is that almost every point $x\in M$ must be recurrent. This is a consequence of the Poincaré recurrence theorem.
Edit: I missed that you are looking at a diffeomorphism. There are no expanding diffeomorphisms of compact manifolds, so the above theorem would not apply.