The way that Bonnie and Clyde's narrative's put together is heavily influenced by a movement called the New Wave, which first became prominent in France in the late 1950's. It used a bunch of film techniques people weren't used to seeing in films—and it created quite a stir.
Let's take a look.
Eliminating Traditional Establishing Shots
What's the first thing you see in this film? Lips.
There's no panning shot of a small town. There's no scene of Clyde walking down a street. We don't see the outside of a house—we don't even see an entire face. Instead, we see Bonnie's lips.
This was wild back in the day… and totally disorienting. It eliminated the idea that setting was vital for understanding what was going on and, instead, moved straight into characterization. In the first seconds of the film, we already understand that Bonnie is beautiful, bored, and sexually frustrated—before we even know what country she lives in.
This lends a whole new meaning to the phrase "up close and personal."
The editing in this movie is justly super-famous. One of the biggest tricks up the editor's sleeve is crosscutting, which allows the camera to show two things happening concurrently—and wasn't being used all that often in 1967.
This spreads on a thick layer of irony. We get to see Bonnie and Clyde finally—finally—having sex at the same time that Malcolm is selling them out to a Texas Ranger. We get to see C.W. parking a car at the same time that B & C are robbing a bank.
We want to hand the mic over to famous film critic Pauline Kael, though. Her insights on Bonnie and Clyde are awesome—in fact, she's one of the people credited for bringing the movie such fame and glory:
The editing of this movie is, however, the best editing in an American movie in a long time, and one may assume that Penn deserves credit for it along with the editor, Dede Allen. It's particularly inventive in the robberies and in the comedy sequence of Blanche running through the police barricades with her kitchen spatula in her hand. […]
The quick panic of Bonnie and Clyde looking at each other's face for the last time is a stunning example of the art of editing. The end of the picture, the rag-doll dance of death as the gun blasts keep the bodies of Bonnie and Clyde in motion, is brilliant. It is a horror that seems to go on for eternity, and yet it doesn't last a second beyond what it should. The audience leaving the theatre is the quietest audience imaginable. (Source)
Yup. Although we didn't see Bonnie and Clyde in the theaters, we were the quietest audience imaginable…as we sat in our couch with our blankets pulled up to our chins.