I've done several approaches in the gym environment, my last one was over 3 years ago, and all but one of them turned into being a very bad situation for me. I'm not saying this to discourage you, just going to give you some input from someone with experiences related to this.
Pretty much all of my approaches I essentially just walked up to them and asked them what their names were and attempted small talk. None of them really gave me any IOIs that I was aware of but they also weren't clearly looking to avoid men. These are pretty obvious, but if she is wearing a baseball cap, always staring at the ground avoiding eye contact, etc. you know to just not bother. I usually did not ask for the number the first time interacting with them and usually just let them go back to working out and would reengage with them the next time I saw them, if they are a regular of any sort you will probably run into them.
Now, here is the bad part. Many women in this environment, especially today with the instagram attention whoring, are quite honestly looking for validation and attention. I had one that approached me after my first interaction with her, we talked about what we liked to do and we were supposed to go hiking and exchanged numbers. She flaked on our date and then ghosted me for a week but when I saw her after she was flirty, etc. Long story short, she was just an attention ***** that did this to tons of men regularly. I had another instance where a woman rejected me and I said "nice to meet you" and walked off. This particular woman went OUT of her way to engage with me, flirt, etc. and basically just tried to lead me on for attention. I had another one that was a bit older than me that was in a relationship and would ask me out but then said she had to mention she was not single, she kept doing this repetitively.
Lesson to be had, if they are flakey, leading you on and not falling right into place for you move on. It's honestly fairly easy to filter these women out. If she is going out of her way to toy with you after rejection or whatever, I'd honestly make it very clear that you don't want her to be interacting with you. Crazy chicks do crazy things lol, just don't take any of it seriously.