I want to hangout with my best friend's ex boyfriend
When did you decide you wanted to hang out with him?
The first time I saw him at my friend's party.
I can understand your feelings on the matter, but you shouldn't get involved with a friend's ex. This will cause trouble between the two of you and you don't want to lose a friend over a boy.
I think I can convince my friend to let me date him.
It is good to talk to your friend. However, you should respect her wishes if she asks you not to pursue her ex.