Try to be natural
I've been diagnosed with BPD too, so I totally understand the whole "I'm trying to be nice and coming off as annoying/creepy/awkward/too-intense".
So, my recommendation, take the pressure off the whole situation to be natural.
Look for a time in which there is no pressure, i.e. waiting in line for lunch, waiting for the teacher to start class.
Approach to her as if you didn't know anything about her. If you haven't had many conversations before, she might be taken aback by the fact that you mysteriously know the topics you have in common, even if your knowing comes from a genuine interest, it might come off as stalking.
The way you state your question may be an additional barrier for you here. Let me explain:
The fact that you are starting off from
I've had issues with her
may be making you come off as trying-too-hard. Try to erase that background and start off from zero. Even if she doesn't notice the difference, you will feel more confident.
- Stay out of personal questions for now, let her set the tone of the conversation for her to feel confident when talking with you. Taking pressure out of the equation, conversations will flow much more naturally for both and that will probably increase trust.
Come up with a topic, not too deep yet, the trust to talk about deep matters comes with time for most people. For example, if you are having lunch, make a witty comment about food. "That seems not-so delicious. Have you tried X?". Try a simple question for a few days, the conversation will grow soon.
Maybe it seems like "How's your day been?" is a simple question, but if you want honest detailed answers, you need her trust first. If she doesn't know where that question came from or she doesn't feel like sharing her feelings/whereabouts, most likely she will opt to answer "fine" or "boring" or something that ends the conversation right away.
You've got this!
Update: how do you get passed the monosyllabic or vague answers.
I would say that the key is to avoid pushing it.
You can try by making more specific questions such as "how did you do on that test?", "Did you feel comfortable during the conference?". If you still get evasive replies, leave it at that, smile and continue whatever you were doing. In another occasion try with a different phrasing.
When someone I don't know very well asks me a question, sometimes I feel exposed (this person whom I know nothing about is inquiring something about me) and that can lead to uncomfortableness. For me, the thing that works best is to show that I'm willing to open and trust by sharing how I feel. For example, "Did you see the new collection on the library? I found this book and really liked it", "I hate assignments like this", "Did you like the conference? I found it interesting". Make it about her but also show that you want her to know you as well.
I hope it's useful : )