If we had to place Bonnie and Clyde into a specific genre, it would fall squarely into gangster/crime, a genre we can trace back to early silent films.
Usually, these types of films involve people who break the law for easy money, get deeper and deeper into their lives of crime, find out they can never again be respectable, and often die violently.
Often, too, the main characters are in love and realize that their love is ultimately doomed. Some excellent examples of gangster/crime films were made in the 1930's and starred the great actor James Cagney. A few of these include The Public Enemy (1931), Angles with Dirty Faces (1938), and Twentieth Century (1939).
And—needless to say—gangster/crime films are still very much with us. In fact, many have been greatly influenced by Bonnie and Clyde, too. A good example of this is the 1994 film by director Oliver Stone, Natural Born Killers… although Bonnie and Clyde's legacy lives on with Pulp Fiction, Reservoir Dogs, and The Drop.