Whats the best way to approach a girl who has been dumped???

Never ever mention or discuss her ex, even if she wants to talk about it

Forget about sex - don't show that you want it

Be a real friend

Be very patient and ... wait 

Don't come on strong. Be patient and take your time. Allow a relationship to develop very slowly. Remember that if she had a long relationship before and was dumped, she is still hurting. Find ways to build up her self-confidence and her self-image. Always be sensitive to her feelings and you might come out a winner.

In my opinion, I think it will only flatter her if you will approach her and show her you are interested . After all like mentioned above she is hurting and she did loose some of her confidence just because someone thought she isn't good enough. There fore approaching her will just boost her confidence back up.  But you should be careful if she just got out the relationship she might be  very  cynical  about guys.  You might want to leave her for a while but if you don't mind being being a "rebound" go for it.

I myself have recently ended my longtime relationship. First and foremost, if you're friends with this girl already, that's really a plus factor. You should always be there for her, show her that you care as a friend and she'd realize herself that you are interested in her as more than a friend, and when that happens, I know you may have had spent a lot more time with her already and may have shown how you can be her man and how you won't do anything to hurt her.

 Aside from that, don't rush things, if she likes you as more than just a friend, she'll show you in her own ways and give you hints whether or not she wants to pursue a relationship with you. Better yet, if she's a really close friend of yours, she'll even tell you straight out that she likes you and appreciates all the time and affection that you have been showing her during one of the lowest points in her life.

Besides, nothing can make us weak in the knees than sweet guys when we're in such a slump.

There are three things you should follow. 

1) Never mention her ex. Even if she herself mentions it out of nowhere, try to get off the subject.

2) I'm sure she's a really cool girl nonetheless, she's a human being. She can't help feeling depressed or hurt. So don't come on strong. If this girl is worth it, don't come on strong because that's the easiest way to lose her. Selfishly go for her when she's hurt.

3) You can ignore all my other advices and try a different approach. Dive straight in. Maybe she's looking for a thoughtful boy to forget about her ex to go out with. 

It's ur decision to paint the road to a successful relationship.

Be friends with her first, don't show any hints that you are actually interested with her. Be her shoulder to lean on and try to tell her that you want to help for her to move on.