This is another very easy achievement; 5000 comes up very quickly, and it should only take you a few laps of the Beginner course to clear the mark. The trick is to get enough time to get there.
To rack up time, complete these tasks: (these figures are all on the Beginner circuit)
Complete a lap: 8 seconds (a lap is at best around 18 seconds)
Overtake another car: 2 seconds (this seems to award every second overtake)
Drift: 1 second
Good drift: 2 seconds
Hit a high speed: 2 seconds
Go through a turn without braking: 2 seconds
Go through a turn without lifting off the accelerator: 2 seconds
Hit a cone: 1 second (this also helps boost the numbers for "100 hits")
That list is not exhaustive, but it will certainly give you enough time to rack up 5000 on the odometer. You don't have to survive the full 80/40/20 laps, as long as your score is over 5000 you will get the achievement.