Don’t know if she even likes me...

There is a girl in my class at college whom I have had a crush on since our first year. We don’t know each other well and have only chatted a few times, but I am sure she knows that I like her. We have common friends, and one of them has told her about me, but she has never said anything or shown any interest towards me. The more I see her, the harder it is for me to get over my feelings, and this has begun to affect my studies. I am thinking of just cutting her out of my life for good and not speaking to her again because it is the only way to move forward. At the same time, I worry that doing this and giving up means I am losing out on the possibility of being with someone I genuinely care about. Should I approach her and be honest one last time, to see how she reacts?
You have feelings for this person but have never taken the time or effort to get to know her. The first step towards trying to find out if someone cares about you is to give them an opportunity to get to know you better. Feelings don’t appear overnight, without context. If you have been in the same college for years, why not ask her out or try and speak to her more often? Treat her as a person you would like to be friends with, rather than someone who has to acknowledge your feelings and respond with a positive or negative response. Start by trying to be a friend before telling someone you care about them. Real life isn’t like a movie, so be realistic about what you want from someone.

How do I know if someone I am dating is gay? I get that feeling but can’t bring myself to ask him without hurting his feelings. 
Sexuality is a private thing, so the person you are dating can choose to be honest or not.