In the new field of financial systemic risk, the network of interbank counterparty relationships can be described as a directed random graph. In "cascade models" of systemic risk, this "skeleton" acts as the medium through which financial contagion is propagated. It has been observed in real networks that such counterparty relationships exhibit negative assortativity, meaning that a bank's counterparties are more likely to have unlike characteristics. This paper introduces and studies a general class of random graphs called the assortative configuration model, parameterized by an arbitrary node-type distribution P and edge-type distribution Q. The first main result is a law of large numbers that says the empirical edge-type distributions converge in probability to Q. The second main result is a formula for the large N asymptotic probability distribution of general graphical objects called "configurations". This formula exhibits a key property called "locally tree-like" that in simpler models is known to imply strong results of percolation theory on the size of large connected clusters. Thus this paper provides the essential foundations needed to prove rigorous percolation bounds and cascade mappings in assortative networks.