You must have completed this track in Championship mode before you can attempt the time trial. The trial consists of 3 laps of the track.
Before you start the time trial make sure you change your cars performance stats in the 'BOX'. You get a certain amount of points to play with. I took 1 point from Braking and Tyres and 2 from Turbo and added 2 each to Speed and Acceleration. I used this set up for every track and didn't really have any problems. If you are struggling maybe think about taking another point from Braking, Tyres and Turbo and adding them to speed or acceleration. It's not necessary but it's something to think about if you're struggling.
Another tip which helped me is to be easy on the throttle, despite this games' arcade style slamming the throttle down every time will cause you to have some tyre spin and lose a lot of speed or crash so bare that in mind.
Here is a video of my attempt it may help you it may not. I suggest maybe looking at where I gain time on you so you know where to make up time if you are coming up short.
Good luck