The next time couple of times you catch him looking your way, give him a BIG smile so he knows you've noticed. If he responds, you know he's sharing your joy. Note that shy people may be embarrassed at first, so they may blush or smile and look down briefly. Don't look away or automatically assume this is disinterest. Over time try to hold his gaze. Develop it into a flirty game. Hold up a sign that says your name or draw a funny picture. Hopefully, he'll do the same thing. After a while of flirting back and forth, you'll both hopefully work up the nerve to talk to one another between classes.
It would also be helpful to be in the same places he happens to be if you can arrange that. Maybe you can trade seats with a classmate so you sit closer to him? Maybe you can join some of the same clubs he's in? Good luck in talking with your shy guy!