If lee sits around autoing for awhile with a bunch of damage items then you have to ask yourself -
what's your opponent doing?
If they're ranged, they're kiting you and you aren't really autoing as much as they are
If they're melee, they're probably CCing you and messing with your spell/auto weaving (maokai/irelia) or bursting you 100-0 immediately (riven)
Most melees don't get to auto attack a lot, reliably. Lee Sin's kit is built around high AD ratios and %execute damage. The autoing with his passive is more for early game and jungle clear. I wouldn't really recommend a build that's based off of autoing because you won't get very many off in team fights if you're building squishy.
I think if you're going to build damage on Lee you need to build burst oriented items like Hydra and Bloodthirster. His abilities scale ridiculously well with the AD while a BotRK and Cleaver will grant very little.