My best friend in the whole world is a chinese guy. Now he lives in Hong Kong (where I just visited him, and got to stay rent free, for 2 weeks!)
I originally met him through class. Yeah, he wanted to "practice his english" and originally I didn't think much about him. I am really close and friendly with many of my students, but I stop short at revealing too much of myself, or talking about my problems. I am always aware that I am their teacher, and therefore there is a distance in our friendship.
But this guy was older than the other students and a different major. Since he wasn't part of the class I didn't have to worry about upsetting the class heirarchy by spending time with him outside of class (I try not to show favorites). Eventually it grew into a real friendship and I would count him as one of the best, best friends I've ever had. I've had some recent troubles in my life and he was the one who helped me through them.
And I agree that having close chinese friends is really rewarding. You DO get to experience and learn about the culture in a totally new way. I got to spend a week in his hometown, a small village, living with his parents, meetings his grandparents, aunts and uncles. It was one of the best experiences I've had since coming to china and he said if I was his girlfriend, it never would have happened. And if I was a chinese girl, it also wouldn't have happened (it would be too inappropriate) but because I'm a foreigner it was no problem and nobody thought anything bad (though all the gossiping locals assumed I was his wife,ha ha ha)
So I forget where you teach, but if it's a uni, don't totally overlook the other students in other majors.