Can we use forcing to construct models of ZFC and ZFC + GCH starting from c.t.m s of ZF? The usual way to obtain the associated relative consistency results (Con ZF implies Con ZFC and Con ZF implies Con ZF + GCH)is to look at the class sized model L (of ZF) and I was interested in seeing if there was an alternate way to do this. Is there any reference material I could look up?
(I do realize that a model of ZF + GCH will model AC so the the question was phrased this way in case it was possible to do the first construction but not the second)
Not always.
There are models of $\sf ZF$ which cannot be extended to models of $\sf ZFC$ without adding ordinals. Since forcing is a technique which does not add ordinals, this means that this is impossible. Note that when I say that, I include class-forcing as well.
Moreover, if you only limit yourself to set forcing, then the answer is an even simpler no. One can easily construct (using class forcing) models of $\sf ZF$ such that no set forcing extension can satisfy the axiom of choice. Similarly one can produce models of $\sf ZFC$ that have no set forcing extension which satisfy $\sf GCH$.
Do note, though, that given a set model of $\sf ZF$ one can produce a set model of $\sf ZFC+GCH$ by simply taking the $L$ of that model, which will be a model of $\sf ZFC$ satisfying $V=L$ as well.