There is actually another way you can do this achievement with just one player (your main account). It's pretty easy so I'll just walk you through it:
-Choose any racetrack and select to be the Turrent
-Move or switch turrents to find a view of the power-ups on the track
-Look for the power-up that shows the turrent with the green "+"
-Shoot the power-up with your turrent
-Upon shooting the power-up, you should be notified that you have aquired a power-up
-Either switch to another view to shoot another power up, or wait for it to respawn (takes a few seconds)
-Repeat until you gain the missles, which is the final power-up
*Blip* Achievement Unlocked
I personally did this so I know it works, you don't need another controller or player(s). You can do this solo against bots.
Hope that helps!