Related to this question, I would like to know, if there is an explicit presentation of the $(\infty,1)$-category of a model category of abelian chain complexes, but this time in terms of simplicial categories. (Categories with some kind of simplicial sets as hom spaces).

The motivation is more or less the same as in the link above,i.e. as a good and understandable example of $(\infty,1)$-categories.