Getting into motorsports is very though. Very tough. I don't want to scare you and destroy your dreams, but you have to be realistic here and willing to sacrifice everything. Especially (as you've already know) you're starting late and have no real experience yet. I am 24 and I have been obsessed about racing since I was 6. Despite having a good chance of getting into racing, I feel I'm not world champion material. Passion helps but it doesn't really do anything until you're actually racing in the first place. If you're not winning in racing there's no point, and that's what made me chose to pursue other options in life. Still love racing though
Basically it comes down to 3 things:
1) Talent
2) Money
3) Connections
If you have talent, you might not need as much money. But even some talented drivers can get shafted by pay drivers these days. You have to be Senna like if you want to get noticed on talent alone.
If you have money, you can generally get by. Just be a safe consistent driver, no need to top the timesheets every session. However, it is unlikely that you will stay in motorsports for long because 1) money will run out without results eventually and 2) there are always other guys bringing in more money than you.
If you have both, you have a good chance. But let's face it if you do you won't be asking for help in a forum, right? If you don't have both, well forget it.
Connections is the final key that will get you through. Knowing people and teams in the business will give you opportunities that might just put you above the next guy when you're both evenly matched. Also having a mentor is invaluable. People that knows the ins and outs of racing, both on and off the track to guide you through the minefield.
Now in your case, if you really want to do it, you want to maximize the 3 factors above:
1) Practice. Practice. Practice. There's no substitute. You HAVE to be driving every single of your waking second, and even in your dreams. Race virtually, race in your head, analyzing your driving even when you're driving on the road, and get some track time. Start with gokarts or autocross, as they are the most accessible. Gran Turismo is an OK starting point, but it's not enough. Get a PC and get the latest simulators which are much more demanding. It's all about having a wide variety of experience in numerous conditions. Every kilometer driven helps.
2) Money you just gotta work, work, work. You might need to get multiple jobs. Get a loan if you have to. Ask as many family/friends as possible to support you. Make a website/facebook/blog/kickstarter or whatever to support your goals. Talk to local businesses and such to see if they are interested in sponsoring you.
3) You need to immerse yourself in the world of motorsports. It's easy these days to just think everything can be done online, but there's no substitute for being actually present on the track and talking face-to-face to the people in the business. Go to your local kart/autocross track and talk to as many people as you can. Go to race meets. Go to car shows. Try marshalling or volunteering at one of those events. If your local University have a Formula SAE team, get involved. Put yourself out there and show that you're interested and damn serious about this.
If you do those things you *might* get somewhere, but you also might not. There are also lots of people wanting to do the same as you that are better. The pyramid of racing is very narrow at the top. If you are really good you might win something like GT Academy and get a shortcut, but recently even GTA winners have lost their seats. Nothing is a guarantee.
If you goal is just to race at Le Mans once, I think getting a ride with one of the gentleman teams are the best bet. Start gokarting/autocross, go to your local club touring car races, then build up to the big leagues (e.g. Blancpain GT3/Ferrari Challenge/Lamborghini Super Trofeo/911 Cup). If you're good hopefully you'll get noticed and one day you get offered a seat to start at Le Mans.
But it's a big IF
If I were you, I'd get off this forum and start going out there right NOW.
Good luck!