The following is a result of Woodin (the proof is found after Theorem 5 of "Generalizations of the Kunen Inconsistency" by J.D.Hamkins, G.Kirmayer and N.L.Perlmutter):
(Woodin) Let $V[G]$ be a set-forcing extension of $V$. Then there is no non-trivial elementary embedding $$j:V[G]\prec V.$$ Corollary 6 gives the theorem from the perspective of the extension as: If $$j:V\prec M$$ is a non-trivial elementary embedding in $V$, then $M$ is not a set-forcing extension of $V$.
From the point-of-view of the generic extension, the corollary can be read as something like "I am not an ultrapower of the ground model by $U$." This must be true for every ultrafilter $U$ in the generic extension. Since the generic extension was an arbitrary set-forcing extension, it seems to me that the corollary implies that ultrapowers of $V$ (or maybe I should say "transitive collapses of ultrapowers of $V$"?) are not obtainable from set forcing (over $V$).
If this is true, I wonder if this was known before the proof presented in the "Generalizations..." paper and even if there is another, substantially different proof (whatever this could mean).
Further (and perhaps this is a silly question with an obvious answer) can ultrapowers of $V$ be obtained by class forcing? Given a transitive set/class $M$, could forcing over $M$ (where the p.o. is considered a class from the point-of-view of $M$) yield a set/class which is isomorphic to some (all?) ultrapower of $M$?
(Added later: I take Woodin's original result to say something like "the ground model is not the transitive collapse of any ultrapower by $U$" is true from the point-of-view of a set-forcing extension's point-of-view. Is this understanding correct?)