First of all - this achievement requires Title Update 1, as TU2 broke it by including 2 unreleased cars to the requirement.
There are 43 cars in total to drive, most of which are available in the Race Day which is going to be the fastest way to get the credit. To make sure it's counted, drive at least 2 miles/kilometers then crash out and retire.
Aston Martin DBR9
Audi R10 TDI
BMW 320SI - Touring Cars
Chevrolet Camaro Concept
Chevrolet Corvette C5-R
Chevrolet Corvette C6-R
Chevrolet Lacetti - Touring Cars
Courage C65
Courage LC70
Creation CA06/H-JUDD
Dodge Challenger Concept
Dodge Charger SRT-8 Drift
Dodge Viper SRT-10
Ford Mustang Boss 302
Honda NSX-R
JRC FJ1000 - Open Wheel
Jupiter Eagleray MK5 - Demolition Derby
Lamborghini Murcielago RGT
Lola B05/40
Lola B06/10
Nissan 350Z
Nissan Skyline GT-R Z-Tune
Panoz Esperante
Plymouth AAR Cuda
Pontiac GTO
Porsche 911 GTR RSR
Saleen S7R
Spyker C8 Spyder
Toyota Corolla GT-S (AE86)
Toyota Soarer
Toyota Supra
TVR Tuscan Challenge
Some of the cars are not, however, you can select the events needed for them in Online multi-player (or grind the World mode, though that will take you MUCH longer):
Dallara Formula 3
Dodge Viper SRT-10
Ford Doran JE4
Ford Mustang GT-R Concept
Koenigsegg CCGT
Koenigsegg CCXR
Mazda 787B
Mazda RX-7 FD3S
Nissan R390 GT-1
Nissan Silvia
Pagani Zonda R
Subaru Impreza
Once you meet the requirement (43 out of 45): clear your drive's cache, remove Content Cache and connect the drive you have TU1 on. Upon crashing out with any car again, the achievement will pop once you're in the loading screen.