Defeat the boss.
The boss is unlocked after 138 cups in the career. This trophy will pop after defeating him. He can be annoying at first but he is fairly easy to beat once you have developed a strategy. Pick a car with a high mass, as the main problem is that he can fling you from the track with ease with his various weapons. Acceleration is not important as all vehicles have the same basic speed. When he is behind you, stick to the edges of the track where his autogun cannot hit you, and focus adjusting your speed to dodge the saws. Then when he is in front, just throw everything you can at him. Missiles cause the greatest damage but autogun is good too. Be careful to dodge his missiles, making sharp turns is effective against these. He will get easier the longer you initially last since the more times you hit him, the more of his weapons you will destroy. Rinse and repeat while he is switching from behind to in front, eventually he gets weaker and weaker until it is just a matter of time before he dies.