You have to beat the story mission "Father/Son" to unlock the car repair shop. You will also need quite a bit of money to fully upgrade a vehicle. Not all vehicles can be modified, but Franklin's default car is perfect for the upgrade. If you would like to keep your money after the achievement/trophy is unlocked, just return to a previous save before you modified a vehicle.?

Now enter the Los Santos Customs (car repair shop) and buy the following upgrades:
1. Armor -- Armor Upgrade 20%
2. Brakes -- Street Brakes
3. Brakes -- Sport Brakes
4. Bumpers -- Custom Front Splitter
5. Bumpers -- Custom Rear Bumper
6. Engine -- EMS Level 1, Level 2, Level 3
7. Exhaust -- Dual Exit Exhaust
8. Hood -- Vented Hood
9. Horn -- Truck Horn
10. Lights -- Xenon Lights
11. Plate -- Blue on White 2 (buy what you don't have)
12. Respray -- Primary Color -- Chrome
13. Repsray -- Secondary Color -- Chrome
14. Skirts -- Custom Skirts
15. Suspension -- Lowered Suspension
16. Transmission -- Street Transmission
17. Turbo -- Turbo Tuning
18. Wheels -- Wheel Type -- SUV -- Benefactor
19. Wheels -- Wheel Color -- Black
20. Wheels -- Wheel Accessories -- Custome Tires
21. Wheels -- Wheel Accessories -- Bulletproof Tires
22. Wheels -- Wheel Accessories -- Black Tire Smoke
23. Windows -- Light Smoke

Once you have all of that stuff, leave the shop to unlock the trophy / achievement. It won't unlock while you are in the shop!

You can also buy Los Santos Customs in Gran Senora Desert as Franklin for 250K to receive unlimited free modifications.