What if it hadn't been self-defense? What if Libby really did just confront Nick and shoot him in cold blood — would she have been protected under the Double Jeopardy clause? No, absolutely not. The clause does protect individuals from being tried twice for the same crime — but that means the same crime, not two separate instances of a criminal act. This is key, because every single lawyer you might ever ask this question will tell you that if Libby had just shot Nick, the act would have constituted a different crime, which happened in a different place, under a different legal jurisdiction — and she would totally have been held criminally liable.
Sure, the movie kind of skirts this issue; after Lehman implies to Nick that Libby (who is currently pointing a gun at him) could just go ahead and murder him with impunity because of the Double Jeopardy clause, Nick pulls a gun, wounding Lehman and shooting at Libby, thereby giving his wife an easy defense. But that's neither here nor there — the movie spends the entirety of its run time with Libby (and the audience) convinced that she'll be completely in the clear if she just blows Nick away, and while this certainly makes for a scintillating premise for a movie (one which sports the tagline "Murder isn't always a crime"), it simply doesn't work this way in real life.
As it turns out, the Fifth Amendment contains no "Killing the Jerk Who Faked His Death and Framed You For the Murder is A-Okay" clause. If you can get past that slight inaccuracy, though, Double Jeopardy is a fun flick, with great performances from Judd and Jones, not to mention a great message: don't ever turn to Hollywood thrillers for legal advice.