In Lurie's HTT, Remark 1.1.4.2. says that: every simplicial category can be regarded as a simplicial object in the category $\textbf{Cat}$. Conversely, a simplicial object of $\textbf{Cat}$ arises from a simplicial category if and only if the underlying simplicial set of objects is constant.
Do you have any hints?
Remark that a simplicial object in $\mathbf{Cat}$ is the same thing as a category internal to $\mathbf{sSet}$. This way, you should see the connection more easily. Indeed, a category internal to $\mathbf{sSet}$ is almost the same thing as a simplicial category except that the objects now forms a space (by which I mean a simplicial set) instead of a mere sets. Lurie's remark is just that sets can be viewed as discrete spaces.
Of course, you should make all that formal. (You only asked for hints, so I didn't go into details.)
Side note : "simplicial category" is kind of ambiguous, even if I agree with the meaning you put behind the name; "category enriched over simplical sets" does not leave any doubt about what you mean.