Issues with coworker!

I'm a recent college grad who has been at my job for about 7 months. I'm 26 & I've had "big girl" jobs in the past, but this job is the first one I've had in my career field. I work for a large nonprofit, and it's a great company to work for. HOWEVER, I have a female coworker I cannot stand and it's affecting me more than I'd like it to. She was employed here for a few years prior to me coming along, but she had a different position within a different component. Now we're "equals" with the same title, same pay, etc. The only difference is that she's in her early thirties and doesn't have a college degree. Anyhow, I've had an issue with her from the very beginning. Every time I would make a suggestion about something, she would say things to dispute my comment. I spoke to her about it & we moved past it. Or so I thought. She comes in every morning, walks right by my open door, and never says good morning unless I say it to her first. She always interrupts me when I'm talking or doesn't allow me to finish what I'm saying. I've observed her with other coworkers and she's as nice as she can be with them. She doesn't come across as "mean" but her behavior towards me is affecting my job performance because we have a small team and our success relies on proper communication. I'm hesitant to speak to her about my feelings because she'll probably deny any wrongdoing and nothing will change. I'm also unsure of how to approach it in general. (I think part of her behavior towards me stems from her insecurities and my strong personality, but I could be wrong.) I usually just avoid her until absolutely necessary. How would you handle it?