There are 16 head to head races in the game. They are signified by the picture (as seen in the achievement tile) of two helmets facing each other. You will be racing one other car, normally over a single lap. Simply win all of these races to unlock this. You will get this while going for "Perfectionist."
Club Division: McKinley Head to Head, Ford Focus Sprint
State Showdown: Mustang GT500 Getaway, Millbridge Duel, Mayapan Jostle,
Grand National: Sonoma Sprint, San Arcana Showdown, Jaguar XKR Duel, Alkeisha Island Dash
Pro Circuit: Balladonia Showdown, Ford GT Head to Head, Belmont Super Duel,
World Series: RSR XKR GT Showdown, Notting Forest Showdown, Mayapan Ultimate Duel, Motul Autech Face-Off