Tell your best friend your dating his sister
Be calm and confident, and above everything else, be a gentleman.If you act like a jerk or treat her badly, your friend will not approve, and you might lose your chance of becoming close to the sister as well as your relationship with your friend.For some context, I met her around a year ago and have gradually developed a pretty severe crush on her.At first, It was just an infatuation, but as I've gotten to know her better I've developed deeper feelings.Tell your friend that you don't want your friendship to change, or for there to be any awkwardness between the three of you.Make your friend feel comfortable enough to tell you how he feels about the situation, and if he does not give his approval, consider letting the prospect of dating his sister go.
Attraction often causes people a lot of problems, including anxiety and insecurity, especially when the object of your attraction seems to be out of your reach.
In that case, I can see problems arising indeed, but maybe the friendship isn't all that important.
I've never really been in your shoes, but I can give you something like your brother's perspective: Are you an asshole? It takes some reflection to realize you are one, and sometimes your friends won't tell you straight up about it.
Tell him that you only have the best intentions with his sister, and that you will treat her with respect.
Be honest with him about how you feel, telling him that you feel something for her that just won't go away.