Accept full responsibility for breaking her heart and apologize sincerely for your actions.
Ask her if she'd like to start again, as friends first, and see where things go.
Listen to her feelings and thoughts and be extra sensitive to her needs. When she asks something of you, be willing to help her out.
Spend as much time with her as she'd like to. This could be a lot or a little; you must respect her wishes and eventually she may want to see you often.
Go out and do things with her that she enjoys, even if you dislike the activity. This will help her see that you are committed to making her happy and winning her back.
Write her an open, honest letter about how much you regret ever hurting her. Give it a lot of thought and be sure to proofread before allowing her to read it.
Ask her out on a date once you've started to get closer. Let her choose the destination and activities for the date and then be sure to pay.
Be willing to make changes in order to accommodate her needs in the relationship. If you have a certain behavior that upsets her, such as lying, do as much as you can to change the behavior or work through it together.
Move slowly with the new relationship, especially if that's what she asks for. She will need more time to rebuild her trust in you before things can get serious again.
Accept it if she says no to dating you again. Remain kind and friendly toward her and she may be willing to give it another try in the future.