First of all, you need to unlock 4 out of the 5 cars as only the DBS is available straight away online.
Aston Martin DB5 is unlocked from the start. Complete 6 events online with it to unlock Dodge Charger R/T.
Complete 6 events in Dodge Charger R/T to unlock Pontiac Firebird T/A
Complete 6 events in Pontiac Firebird T/A to unlock Aston Martin DBS
Complete 6 events in Aston Martin DBS to unlock Shelby GT500. Total of 24 events.
You can either do this normally online and get some mileage in as you will need it anyway, or you can grab a friend and do some quick events over and over again. A good event for this is located on the SW corner of downtown, a challenge called "Pyramid parking". Check the map at the end of this guide for the exact location.
In order to unlock the "Movie fan" license plate needed for Movies: Fan Friction you will need to unlock every single non-dlc standard mod for all five cars. Every car will come with Burn Nitrous already unlocked. Track tires, Reinflates, Reinforced chassis, Short gears and Powershot nitrous will unlock via mileage. While the order varies, the last of them will always unlock upon reaching 200 miles driven. The last 4 mods require specific tasks with specific upgrades equipped:
Impact protection body requires 10 takedowns with powershot equipped.
Long gears requires 5000 yards of drifting with track tires equipped.
Off-road tires unlock upon getting a total of 1500 yards of air with reinforced chassis equipped.
Aero body requires 60 near misses with short gears equipped.
All of these can be done in a private lobby by yourself besides impact protection body. At least Aero body and long gears should unlock naturally if you unlock and equip the related mod at 100 miles mark or before.
There are few approaches to reach the 200 miles in each car:
1) Just play normally on speedlists, takes longer but if you like the game online then why not do so? If you do it like this, add 9 hours to the estimated time.
2) Drive round the main highway repeatedly. If you do it counterclockwise, there are less sharp turns and you can maintain a higher average speed. This is the quickest way but also the most boring way as well.
3) Find a suitable edge that will respawn your car upon driving down so that it's heading to the edge again. Tape R2 down. This is propably the least fun way but requires nearly zero effort. Add about 15 hours to the estimated time.
Regardless of how you unlock the plate, you will now need 5 slam takedowns with the plate unlocked. A slam takedown is a takedown from the side, by slamming your own car sidewards into a victim. The difference between your speeds can't be too much or it won't count.
The trophy description is slightly misleading. What it doesn't tell you is that you need to get the takedowns in a private lobby. Thanks for ChrisCSW20 and Naps for help in figuring out the exact criteria.