Reference textbook developing NBG set theory I'm starting Borceux "Handbook of Categorical Algebra". It starts with a brief discussion of the logical foundations of category theory.
He describes two approaches: 1.defining universes and 2. With NBG set theory.
I'm interested in the second approach, so I want to study the axiomatics of that system. 
The book that I'm looking for should present the logical construction of the theory and some important results such as proving that NBG is an extension of ZFC, some results on model theory and a profound discussion of its applicability to category theory if possible.

Mendelson's Introduction to Mathematical Logic has a good development of the basic facts of set theory in NBG and includes a discussion of its relationship with ZFC (it is a conservative extension of ZF, and hence the model theory of NBG is essentially the same as that of ZF). As for a profound discussion of its applicability to category theory, I don't know of one. The system MK introduced in the appendix to Kelley's book on topology is probably closer to ordinary practice in category theory, since it is common to quantify over all elements of a proper class in categorical constructions. Both systems will let you talk about particular large categories, e.g., topological spaces or groups, and about particular functors between them, e.g., the fundamental group functor. Neither system allows you to talk very satisfactorily about large categories in general.
