First you join the army. Then you make it clear that you'd like to be a sniper. Then after you learn all the basics, they will put you through a series of very difficult tests. If you pass them all, and they think you are the right person for it, then you might be able to become one. Otherwise, they will make you do some other career they think you are better at.
The problem is that when you join the armed forces, you are handing your life over to them. You can't just quit if they give you a job you don't want.
I've read an interesting article, written by a real sniper in the British army.
Being a sniper is more than just being a good shot. You have to be able to do complex math in your head. Account for temperature, the coriolis effect, wind speed, wind direction, bullet trajectory, and much more.
You might have to camp out in enemy territory for days on end, living off your supplies. And then when the time is right you have to look through the sights of that gun and kill another human being. The sniper who wrote the article I read said that sometimes under certain situations he must shoot a person in the head. Sometimes you get to see the expression on their face as they die.
Quite honestly, there are not that many people who can do that kind of thing and live with themselves. You are talking about having a front row seat to a violent murder. You may be following orders, but it is still you pulling the trigger.
I suspect if you arrive at an army recruitment office fresh out of school, claiming you want to be a sniper, they may not take you that seriously. You probably need to set your sights a little lower to begin with.