How do I know whether she's the person I should marry?
Other than her willingness, it's a combination of things — a little bit of art and a little bit of science. By that I mean God will likely use two sources through which to communicate to you about such a decision: your heart and your head.
Your head helps you address the fairly obvious things, like:
These are the kinds of questions a couple might encounter in a pre-engagement class, which are becoming increasingly more popular, and from my view, advisable. I realize that relationship survey questions are about as romantic as, well, a survey, but a little planning can go a long way to helping a lifelong romance. If you're serious enough to be discussing marriage with her, a formal way of addressing these important topics needs to be in the mix.
Now a few thoughts on listening to your "heart." By "heart" I'm not just referring to how you feel about her emotionally. I assume you have strong feelings for her or we wouldn't be having this dialogue. What I mean by "heart" is that intangible "peace" that God gives us when our life or our individual decisions are moving in accordance with His will. Paul describes it as a feeling that "transcends understanding." The more we engage the Scriptures and commune with God, the more sensitive we are to His leadership, often in the form of that inward "peace."
Pray over your relationship. Pray for God's will to be done in her life and in yours, no matter what it is. And pray that you would be sensitive to His voice. He will lead you in the right direction. And if you're going the wrong direction, he will let you know. Don't ignore the red flags (or even yellow flags) that He may place in your way.
As for my own experience, I don't recall wondering whether my wife-to-be was the person I should marry; all I knew is that I wanted her to be! I was crazy about her. The more I got to know her, the more painful it was to be apart from her. It wasn't long into our relationship that I realized how difficult it would be to picture my life without her. To put it simply, when we were together, peace. When we were apart, no peace. Fortunately — and this is key — she agreed, and the rest is, well, you know.