This achievment is tricky because it comes with so many conditions, it's not actually that hard.
To do this you have to select Arcade.
But it does not matter what length of game you select. I recommend a 10 minute game (Or even 5 minute in this case) and once you have reached your target stop.
The achievment is not stackable. It you get an S licence on your first go you will not get the achievments for getting an A, B or C licence. So you will have to play this at least 4 times.
Just to make sure you don't screw up in the last second because a car got in your way to make this a cake walk from the main menu go to 'Help & Options' then to 'Settings' Then change the Time and Traffic difficulty all the way down and you will breeze through this.
To see what you need here are the scores required:
S - $5000 >
A - $4000 > 4999
B - $3000 > 3999
C - $2000 > 2999