What should I do if my ex (we dated for two years and she dumped me a month ago) tries to becomes friends with me again?
If you're still feeling that negativity toward the guy she's dating now, then it's probably way too soon to try to be friends with her. Before one can have a healthy friendship with a person they were intimate with for such a long time, there needs to be a healthy disconnected period where both parties move on with their lives and "get over" the loss of the other. This period can vary depending on the person, but it has been known to take a while (months, even years). In my experience, I've never been able to be close friends with my exes. Even if we ended our relationship on good terms, it's awkward to go back to just a regular friendship. Even now, being married and years after my last breakup, I am still only, at the most, casual "Facebook" friends with a couple of my exes, and we do no more than leave a comment occasionally on a random status.
Honestly, unless your ex and yourself share a very similar sphere of influence and you see each other a lot through your normal routine, it's going to be difficult to maintain any kind of healthy friendship to begin with. Forcing it only leads to more awkwardness.
If you really want to be friends with this girl again, you need to wait until you no longer harbor any feelings of negativity or resentment, toward her OR the guy she's dating. This means cutting off contact COMPLETELY for a while and focusing on other things in your life until those feelings disappear. That being said, and based on what you've told me about the circumstances and her behavior, I don't think this is a person you really want to be friends with. Having a deep relationship with someone for two years, then just dumping them and jumping into another relationship less than a month later is kind of a dick move, and kind of tells me she gave up caring about you long before you two broke up. Her effort to be friends with you again so soon seems like just shallow attempts to placate you and make herself feel less guilty for what she did. It sounds like "charity" to me, something you don't need any part of.
I'd expect you to take my advice with a grain of salt, of course. I have only limited information about your situation, and I am basing my advice off my own experiences and the experiences of people I know and have spoken to about. I hope that you find a resolution to this situation that benefits you and allows you to move forward with your life in a healthy way.