But instead of directly raising this matter, what I had thought of was to ask her: "What would be the best way I can get to know you better?" , which I feel sounds more positive in terms of intentions. Of course most likely she could figure that I'm asking because of her non-replies but,

No no no no no, you don't ask girls like that la

Y not just ask her out for some hiking activity as a start. If she said no without any other suggestion, that would give you a lot of hints already that she's not interested in you.

If she said yes, then you can update us.

In the meanwhile, go get to know other girls rather than focusing on her alone.