First of all...welcome to the EC! I hope you have found a place where you can be real
What @sensi-sante mentioned about being a cancer in the family is oh so true! Toxic people will create toxic atmospheres any place they go. Like an energy vampire, they will suck the good energy out of everyone and leave nothing but hurt in their path. Your empathness had picked up on this...hence your dislike for her. It is sad that we can be an early warning system but one that no one seems to heed advice from. Below is a link that will perhaps give you some pointers on how to proceed in this touchy situation
Additionally, if you don't do it yet, I suggest that you get grounded and centered. Shield yourself as well to prevent her garbage from seeping into you (which sounds like it already has) There are a lot of different techniques to doing this. Check out Elise's Empath Survival Guide for starters
updated by @cat-whisperer: 05/13/17 06:59:01AM
This almost sounds like a narcissistic relationship... I wouldn't know unless I saw it but if she's bringing up problems about trust, she's obviously projecting the worst side of herself on someone else. I know how hard this is going to be before I even recommend it so I understand if you don't like what I'm going to say, but I believe she was put in your life for a reason. It sounds like she's broken and confused with her life, and as hard as it will be, I suggest allowing her to get close to you. Keep your guard up, and keep the focus on her, not you (tell her small details but don't get too personal if you can help it). Focus on her and try to set aside your feelings for her. Just let her talk to you about her life and things and see what she really needs. Everyone acts the way they do because of their past, most people don't know how to let go and live openly without restraints from guilt or things they haven't moved past. Just slowly get closer, you'll find her motive, but don't do the easy things and use it against her, help her out and be someone she can talk to. You have an amazing chance to do some good for someone, here.
Now on the other hand, if it gets worse, don't be toxic towards her and always be kind, but definitely keep a barrier up. Don't "avoid" her, but when she talks and you feel like she's lying, ask about it. Let her know you see right through it. But don't pry around if you choose this route. You'll end up hurting someone else and that is always the wrong thing to do, no matter who you're working against. Stay true and stay positive, and if she chooses to stay on a darker path of selfishness and lying, then eventually she will feel unsafe there. But for the record I think this is a chance for you to help someone else and really build yourself in the process.
Anyway you choose best of luck and blessings to you!