After dating for a year and a half, I had a month of suicidal depression, and we broke up because of the strain that put on our relationship. I have completed NC, improved my life and gotten over the depression with help, so I want to get back together. However, I was so needy and clingy right after the breakup that she told me not to contact her again.
She also got a new boyfriend only a month or two after we broke up. Our personalities fit so well together, and the only fundamental problem with our relationship is gone, but how do I show her how great of a person I’ve become if she’s completely ignoring me? Especially since she’s probably in a rebound relationship.
Talk to her, but every time you do, pretend her new boyfriend will read it. Chances are that he won’t read it if she doesn’t want him to, but it’s a good way of staying within the boundaries. Don’t talk to her that often, just a quick drop in.
When I was with a guy, I couldn’t stop thinking about my ex, and the main reason I didn’t eventually return to that ex is because when he was dating another girl (and I, dating the other guy) he kept on approaching me in a way that sounded like we were still together. What’s important with ex’s, is to maintain distance.
She probably will break up with him and you can approach her about your relationship after that, but in the meantime, keep your head down and be patient. 
Thanks Kimmy123, it’s really nice to have the perspective from someone on the other side of the situation. I wanted to add to my post that I sent a handwritten letter after a month of NC - I didn’t initiate the NC, unfortunately - apologizing for my behavior after the breakup and telling her some things that are going great with me. However, I didn’t apologize for how terrified she must have felt that I was suicidal. I didn’t want to bring up bad feelings, but I wonder if showing that I understand is what she wants…
A few weeks later (also a couple days ago) I sent her a friendly text, but she didn’t respond to that or the letter (I’m not surprised). I’m willing to wait for her and work towards a relationship, but I’m not sure whether I should keep texting her, or if waiting will actually make her miss me. I also think she maybe doesn’t read anything I send her. I really have no idea how she actually feels about me, but I’m sure she’s trying her best to forget about me. We were extremely serious while dating, but might she actually be successful and stop loving me?
P.S. I think she probably will break up with him in the next few months, but I always worry that they might never break up. My main concern is whether she reads anything I send to her - if she doesn’t, I don’t know whether I should text her at all 
P. P. S. Soon after she e-mailed me never to contact her since she was “a very different person now” and has “moved on,” her boyfriend called me on her phone, telling me to stop calling her (I called her twice to arrange for getting back some things I let her borrow, but she only picked up the first time). I assume she told him to do it, so is she trying to make me jealous, or did she just get pleasure from seeing her boyfriend fighting for her and being the dominant male?