Set forcing and ultrapowers 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?)
 A: In section 2.3 of Paul Larson's book, The Stationary Tower, he shows that if $V$ has a proper class of completely Jonsson cardinals, then forcing with the class-sized stationary tower $\mathbb{P}_\infty$ yields an elementary embedding $j : V \to V[G]$.  Here, $V[G]$ is a direct limit of generic ultrapowers.  He notes that $V$ and $G$ are not definable classes in $V[G]$, so this embedding only exists "from the outside."
A: The paper you refer to is available here: Generalizations of the
Kunen inconsistency.
In your question, it seems that things got switched around a
little when you went from the theorem to the corollary, since in
the paper Corollary 6 asserts that if $j:V\to M$ is an elementary
embedding, then $V$ is not a forcing extension of $M$. (But your
question asserts that $M$ is not a forcing extension of $V$.) So
this corollary is simply an equivalent formulation of Woodin's
theorem that there is no $j:V[G] \to V$, expressed from the
perspective of the forcing extension rather than from the ground
model. Theorem 5 and corollary 6 are equivalent ways of expressing
the same fact.
Nevertheless, your formulation of the corollary happens to be a
true statement, because this is a similar re-casting of theorem 7
of the paper, which asserts that there is no nontrivial elementary
embbedding $j:V\to V[G]$. That is, what we've got is no nontrivial
$j:V\to V[G]$ and also no nontrivial $j:V[G]\to V$. From the
perspective of the extension, what this says is that if $j:V\to
M$, then $M$ is not a forcing extension of $V$, as you say.
Both of these results are unified by the claim (theorem 8) that
there is no nontrivial elementary embedding $j:M\to N$ whenever
$M$ and $N$ are grounds of $V$. And this fact is a consequence of
the more general claim in the paper that there is no $j:M\to N$
whenever $M$ and $N$ are eventually stationary correct. This is
theorem 10 of the paper, and it provides a partial answer to your
question about class forcing, since many of the usual class
forcings that one might undertake are eventually stationary
correct.
You ask about the history, and this is a little less clear. This
is what we say in the paper:


Attribution for this next theorem [theorem 7] is not clear to us.
    Woodin reportedly proved it along with theorem 5 while he was a
    graduate student in the early 1980s. But also, Matt Foreman
    mentioned to the ﬁrst author [Hamkins] that he discussed a version
    of the theorem with Mack Stanley and Sy Friedman around the same
    time, but their proof was evidently different than ours, and
    unfortunately the result was not published. Suzuki proved a
    theorem implying our theorem 7 in [Suz98, p. 344], using a
    technique essentially the same as ours. (Suzuki proved that if
    there is $j : V \to M$ in $V[G]$, then $V\not\subset M$. This
    result is stronger than our theorem 7, but weaker than our theorem
    10. Although Suzuki states in his introduction that his proof only
    concerns definable $j$, in fact his proof never uses that fact and
    can be formalized in NGBC.)


I don't know the Foreman/Friedman proof, but this may be the
alternative proof you seek. In any case, it seems that Suzuki may
have the first proof of that fact to appear in print.
