So we all know Bane, Batman Forever, big stupid moron created out of a small stupid skinny man.
So we all know Bane, smart, hulking, big huge guy who break's bat's back?
So we all will know Bane who is an intelligent, terrorist who shoots down pregnant women while working with the League of Shadows.
Lets have a better look at Bane's history: After leaving Gotham post-Knightfall, Bane impressed Ra's al-Ghul who was looking for an heir to carry on his legacy after Batman duped him. Bane accepted, but after Ra's betrayed him, Bane destroyed all the Lazarus Pits in the world.
Now lets look at the Dark Decker Rises and the League of Shadows. Ra's picks out Bane to be his heir, then Ra's meets Bruce and trains him personally, seeing the potential that Bruce has. Ra's wishes for Bruce to succeed him, Bruce saves him. Years later Ra's attacks Gotham and easily bests Bruce in combat, but Bruce tells Ra's that he wont kill him, but wont save him [that can't be how it works... or I would have tried it by now... ON YOU!]. Bane takes advantage and viciously seizes control of the League.
That is my guess on Bane's connection to the League of Shadows.