Co-op mode must be done over Xbox Live as Split Screen does not allow it.
For this, please reffer to This Thread for a Co-op partner.
You will either need to go through 2 carrer modes so both of you can get this and "Co-op Drivers' World Champion" or you can use the method discribed on the Roadmap by Maka below:
"Both players should trade wins back and forth to keep the points very close to each other. Now, on the last race, have the host quit out, and the other player will finish the race to get the achievement. Once this is done, invite the other player back in, and let the host finish the race in first, making sure to win by enough points to beat the other human player."