Yes you should tell her. If she was still in love with him or if they had an ugly break-up, you should expect that your best friend might be hurt by you dating him. After all there are millions of other people that you could date, but you glommed on to this particular guy, why?
Also, it’s probably going to be painful and or awkward if you are ever around the dude and your best friend at the same time. I wouldn’t be surprised if the best friend was so hurt that she drops you as a friend. Even if you didn’t mean to hurt her on purpose, your dating her ex is still going to have the same effect on her (meaning it will make her feel crappy).
Plus, it sounds like you are young. This dude isn’t going to stick around. They never do. And then you will have likely lost or at least hurt your best friend, and then you’ll be heart broken. Hopefully no one YOU know well, will start dating him at that point.
Guys are not worth hurting your best friend over.