Well, your ex is going to be really hurt but I don't really understand why. I mean this person is another person and they have the choice to go out with you or not. I mean, you two are broken up so why should she care? I may be wrong but this is my opinion. You can also talk to your ex or her best friend. Just make sure you're going out with her for the right reasons. Not to make your ex jealous.
Theres really no reason to not go out with your ex's bff, I wouldnt but if that's what you both want to do, then do it, but remember your ex know every thing about you and she can make thing impossible for you during that relationship..