This is really a comment on Hamkins answer, but I'm not permitted to make comments so I'll write it as an answer.
Using the standard Schoenfield machinery for forcing, there is no need for a well founded model. The theory of forcing is developed entirely inside the model $M$, including the definition of the class of names and the forcing relation. This uses the axiom of foundation inside $M$, but not external well foundedness. Given a externally defined generic set $G$, then, $M[G]$ is just the set of equivalence classes of names under the relation $\dot x \equiv \dot y \iff \exists p\in G ( M\vDash p\vdash \dot x = \dot y)$.
This is, of course, essentially equivalent to Hamkins' answer.