Prove You're Capable of Change
Chances are, before you broke up, there were warning signs you'd been neglecting. Maybe she was dropping plentiful hints about how you were not affectionate or romantic enough. Or, maybe she constantly complained about how you never took her out and spent too much time with your friends. It may feel like it, but it's unlikely this thing came out of left field.
Regardless of your perceived shortcoming, you need to take the time to show her you're addressing it. If she broke up with you because she thought you weren't ambitious enough, tell her about the jobs you've applied for in her absence and how well the interviews have gone. If you had an issue with honesty, tell her you've given serious thought to your behavior and the way it must've affected her, and you have a new outlook on the issue.
Your plan of attack will change depending on what the particular issue was, but the point is, you need prove that if she were to actually give you another chance, things would be different.