To own a Most Wanted car, you first need to earn the right to race against a Most Wanted racer by earning Speed Points. You collect SP by doing just about anything. Some tasks that will earn you SP include completing events, doing stunts, competing in autolog, and escaping pursuits.
The number of Speed Points required to compete against each Most Wanted car are as follows:
Most Wanted #10 - 60,000 SP Alfa Romeo 4C Concept
Most Wanted #9 - 135,000 SP Shelby COBRA 427®
Most Wanted #8 - 220,000 SP Mercedes-Benz SL 65 AMG
Most Wanted #7 - 315,000 SP Lexus LFA
Most Wanted #6 - 415,000 SP McLaren MP4-12C
Most Wanted #5 - 520,000 SP Porsche 918 Spyder Concept
Most Wanted #4 - 630,000 SP Lamborghini Aventador
Most Wanted #3 - 750,000 SP Bugatti Veyron Super Sport
Most Wanted #2 - 875,000 SP Pagani Huayra
Most Wanted #1 - 1,000,000 SP Koenigsegg Agera R
Once you've earned enough speed points, you can challenge the Most Wanted racer in a race event. Police will also be in hot pursuit the whole time. Any car can be used for Most Wanted events. Police presence will be greater the further you move up the list. When you start getting higher up, it's a very good idea to have re-inflatable tires equipped to get through all the spike strips.
After you've won the event, you'll be taken back to free roam and be tasked with shutting them down to take their car. Get up close and ram them into a wall to take them down. Each Most Wanted driver you take down will also unlock a corresponding achievement for that particular car.
Don't sweat it if you lose track of them. Just keep driving around and they'll eventually reappear and pull just ahead of you.