Mods, in multiplayer, are unlocked by doing milestones for any vehicle. Each vehicle has many milestones (Drive X amount of miles or have X amount of take-downs).
So anyway, here's what you do.: Hit "Right" on the D pad to bring up the in-game menu. Go to Multiplayer and create a friend game. Now the car I started with was the Ford Focus RS500, but I'm not sure if you will. The first milestone for that car is simply drive 5 miles. All driving milestones can be done in your own game, but for take-downs you will need others (obviously). So simply drive 5 miles in the Ford Focus RS500 and it will pop.
Also, when you get your first mod, you can upgrade your car and see the rest of the milestones. Even the ones needed to unlock pro mods, but that's a separate achievement.