If $\mathcal{C}$ is a category, then surely the category of simplicial objects $s\mathcal{C}$ is not automatically a model category. What conditions must $\mathcal{C}$ satisfy in order for $s\mathcal{C}$ to have a reasonable model structure?

It always has a model structure using Kan's theory of Reedy categories. For a proof, see Hirschhorn Model Categories and their Localizations 15.3. This is because $\Delta$ and $\Delta^{op}$ are both Reedy categories. I will address the more general question as well: If $C$ is any small category, the condition for $M^C$ to be a model category is that $M$ be cofibrantly generated. However, it is not necessarily true that $M^C$ is itself cofibrantly generated. In general, the condition we need for that is called combinatoriality. However, the existence of such a cofibrantly generated model structure that is not combinatorial would disprove Vopenka's principle (a large cardinal axiom), so I cannot give an example of one. 


Quillen gives a couple of sets of sufficient conditions for $s\mathcal{C}$ to be a model category, in his Homotopical Algebra book. Notably, this includes the case when $\mathcal{C}$ is a complete and cocomplete category with a small set of projective generators. This gives model structures for categories of simplicial groups, simplicial rings, simplicial lie algebras, etc. 

