I'm going to date my best friend's ex-boyfriend, because they didn't even go out for a month
Why would you want to do that?
I heard that he is fun to hang out with, and he is so hot!
It's usually a bad idea to have a relationship with your best friend's ex, as this will make things awkward when you are all together. You shouldn't assume that your friend is okay with this, but if she has actually encouraged it, then you can go ahead.
He likes me back and it's not weird, so it's okay
I'm glad to hear that your friend's ex is interested in you, and I think that you both could have a good time together.  However, I have to stress that you should make sure that your best friend is okay with the arrangement as well.  As long as they broke up amiably, then everything should be fine.