As well as having a Driver Level in this game, there is also a Car Level. This level is less important, but is vital for earning this achievement.
I found the easiest way to achieve this is by taking a car you use a lot, and using it in the Championship races. In my case, I used the 2009 Ford Focus RS, as I found it was a suitable car for the job. Upgrade it when necessary, and use it in as many events as it is eligible for. Once you get it to Level 5, you do not have to keep using it.