You get multiplier's by doing well during songs. If you start singing well, you will get a 2x multiplier and if you keep singing well while that multiplier is up, the multiplier will increase to up to 4x, the highest multiplier in the game. If you mess up during the multiplier, the multiplier will reset. You will also get a multiplier after the song is over and it will be anywhere from 1-4. If you don't get a 4x multiplier during the song, but you get it after the song during the Score screen you will still unlock this achievement.
Also, when you get a 2x multiplier the notes will turn red. When you get a 3x multiplier the notes will turn orange. Finally, when you get a 4x multiplier, the notes will turn to gold. That's one way you know when you get a multiplier.