Are there collections of sets that are neither a set nor a definable proper class? Working in ZFC, are there collections of sets that are neither a set nor a definable proper class?
I mean if some collection of sets is not a set can we necessarily conclude that it is a definable proper class?  
 A: This question makes sense when working "outside" a model of $\sf ZFC$. In which case the answer is almost trivially "yes".
Before getting into that, just note that working inside the model there are only sets, and formulas from the meta-theory which you can use to "approximate" collections which are not sets. So if you want to get outside this context you need to either work in "a larger universe" or in something which substantially augments $\sf ZFC$.
If you have a model of $\sf ZFC$, it is a set, and it has only "so many" [read: as the cardinality of the model] classes, which are definable (with parameters), but Cantor's theorem says that there are more subsets than that. So these give you an example.
Specific, and perhaps interesting, examples, are if you have a model $M$ and $N$ is a generic extension of $M$, then there is some $P\in M$ and $G\in N\setminus M$ such that $G\subseteq P$. This gives a "subcollection" of $M$ which is neither a class nor an element of $M$.
Another such example could be non-well founded models, where you cannot "locate" the well-founded part. It is a subset of some ordinal of the model, so it is a collection of ordinals of our model, but it is neither a set there nor a definable class (since it is an inductive class of ordinals which is not all the ordinals of the model, it cannot be a definable class there).
