I've tried BOTRK for a while and I have to say unless I'm losing I'll stick with my Bloodthirster. Trying to get 3 shots in during laning is not worth it and too often it puts me in a bad position or opens me up to harass. Bloodthirster lets me poke with tumble a lot easier and from there I tend to rush a phantom dancer. BOTRK I'll go for when I'm losing my lane and can use the active for defense.
That's standard. BotRK is in a very sweet spot right now do its useful stats, OP passive, and its ridiculous active.
Do -not- rush straight into an IE after BT. Vayne wants more AS. You aren't playing Ashe. --- She's a few cards short of a full deck: a joker in the game, ooh. She's got a bullet with your name on it - no doubt she's a mental case.