How do you become a racing driver?

First, have plenty of money. Then you can learn and practice karting (where all recent F1 drivers (and most other racing drivers) learned their trade). Then, after success, talk to one of the teams that covers most formulae (like Redbull and McLaren) and ask to join their trainee programme. 

If you start now you might get there in around 7 years. There may be other routes into (say) US or non open-wheel racing. Someone else may be able to advise.

If you have a local race track that's where you start, build or purchase a cheap stock car to see if this is for you, and work yourself up thru the ranks. If youre good enough sponsors and the bigger teams will find you and teach you what you don't know. If this is your dream, then find a way to get started.  I got into stock car as a High School Auto Mechanics/Welding class project and it took off from there and made my way up thru the ranks but then back in the 70s, this was much easier than today. 

The biggest thing if you really want to be a pro driver, then be willing to take a risk but if this is your biggest dream then be willing to risk it all but winning isn't everything here but being steady and never quitting. There will be setbacks like hitting a wall or rolling over but its how you get back up and on the track will show if you have what it takes or not.  Good luck