It seems to have been popularized by the television show Babylon 5, which used "The Exercise of Vital Powers" as the title of one of its fourth season episodes (written by creator & executive producer J. Michael Straczynski, who has gone on to get a reputation as a writer of terrible Marvel comics). In that episode one of the characters says that the "ancient Greeks" defined happiness as "the exercise of vital powers, along lines of excellence, in a life affording them scope".
Fine. But where does this come from?*
It is most often attributed to Aristotle -- sometimes specifically to the Nicomachean Ethics. But while Aristotle says arguably similar things, he doesn't quite say that.
So who did say it first?
The answer was uncovered by the researches of Jeffq at Wikipedia (upon which most of this post is based).
Jeffq tried to track down the quote in reference to the B5 episode. He noted that the phrase has become popularized in commencement addresses and the like, often cited to Aristotle... but that all of these seem to post-date the B5 episode.
But he did track down what seems to me the ur-source of the quote, namely, Edith Hamilton's 1930 book The Greek Way. (Jeffq cites a 1964 paperback, but the book was from 1930). In her second chapter, she writes:
The exercise of vital powers, along lines of excellence, in a life affording them scope" is an old Greek definition of happiness. Through all Greek history that spirit of life abounding moves. (p. 24 of the Google Books-available version)A number of later writers explicitly cited this to Edith Hamilton; so far as I've seen, no one has found an earlier citation for it. (If anyone has one, please leave it in comments!)
My guess, however, is that the quote became more widespread due to its use by John F. Kennedy. In this 1963 speech (or proclamation or whatever it was) he wrote that "Happiness, as defined by the Greeks, is 'the exercise of vital powers along lines of excellence in a life affording them scope.'" Presumably he got it from Hamilton, who was a widely read popularist in her day, but he doesn't cite her. Anyway, he may have helped spread the quote around.**
Whoever said it, it's a nice phrase. But until an earlier source appears, it looks like Edith Hamilton, not Aristotle, deserves the credit for this one.
* In a later internet posting, JMS rather unhelpfully said (scroll down) that "It's not so much a quote as the Greek definition of happiness." This rather thoroughly begs the question, since definitions don't write themselves: even if you want to argue that the definition was implicit in an entire culture's view of happiness (which does seem to be what Hamilton is saying, see above), someone needed to be the first to use that phrasing. So of course it's a quote, even if the source of the quote intends it as a definition.
** Actually the Kennedy brothers seem to have been a good conduit for this sort of thing. The spurious ancient Chinese curse, "may you live in interesting times" -- whose earliest known source is a 1950 SF story by Eric Frank Russell -- seems to have been put in widespread circulation by a 1966 speech by Robert Kennedy. (For more on the history of this fascinating spurious quotation, see this Wikipedia page; this Kevin Drum post; and this Language Log post. All of these links trace to Stephen DeLong's research, which is preserved here.)