Since you ask for a duality between quotients of $C(X)$ and the closed subsets of $X$, the following does not answer your question directly. Nevertheless, I hope it might be of interest. I am a tireless proponent of the thesis that if you want to extend the duality between compact spaces and algebra of continuous functions thereon, then the appropriate context is the space $C^b(X)$ of bounded, continuous functions, not with the norm but with the strict topoogy. This was introduced by Buck for functions on locally compact spaces and then extended, using different methods, to completely regular ones. Many of the results for Gelfand-Naimark theory for compact spaces can be carried over in the natural way, in particular, precisely the duality between closed subsets of $X$ and quotient algebras with respect to ideals---one need only demand that the ideal be closed in the above topology. This theory is presented in the book "Saks spaces and applications to functional analysis".