You need to win all 28 Domination races in the Shatter Bay Career. In Domination events the key to winning the race is using your frags effectively whether this os fragging all of your opponents to first place or using your frags to reveal as many shortcuts as possible by smashing through the city targets.
In most cases you will learn to master both of these features simultaneously to succeed, and once you have gotten used to of the games handling and rather unforgiving AI, you should have no problem winning all the races.
Although the races themselves do not tend to get any harder as you progress, the AI is quite tough in this game throughout from the very start and if you make too many mistakes/crash/get fragged too much in an event and you fall behind the pack then you will have a hard time catching up as the AI does not slow down for you, and this is why practice, persistence and patience is required to win all the domination races.
The best car for each car class is the Hurricana CX (Super class), Dorean LE (Race class) & the Motega XR (Street class) but you can still experiment to find the car that best suits you for each class.