You can't convince any girl into relationship - she's got to have the feelings for you, chemistry etc.
How to convince a girl to start a relationship?
Basically you don't, as we can't make anyone feel the same way about us as we do them. But when you find a girl that likes you the rest will follow. You don't force any female into anything they don't want which you best learn, and remember always.
I am a girl myself, and I believe that you should let her know that you like her by subtle movements, like flirting. Get to know her first, what she likes, ect., and then don't ask her out until she returns the gesture, or you, her friends or your friends believe that she likes you back. Also, go slowly because girls hate it when a relationship goes too fast. MOST IMPORTANT POINT- be yourself. Girls find it very annoying when boys try to pretend to be someone else.