Kayn: What am I doing wrong?

I've got Kayn as soon as it was released. For the time passed I have at least a match per day as Kayn. And I still can't understand... where his damage is??? How to really influence game with this champ? Where's his power spike? What is it I am building wrong? Could you advice me on this please? I tried building Black Cleaver {{item:3071}} . I tried Trinity {{item:3078}} to have 20% CDR, movement and Sheen passive on hit. I tried going tanky this way {{item:3025}} {{item:3742}} I tried building {{item:3812}} for AD and additional LifeSteal. I even tried going {{item:3074}} The central problem is... I can't deal enough damage with Abilities even with 40% CDR and enough AD stacked. Going full combo: Q -> aa -> W -> aa -> E-> aa -> Q -> aa -> R --- doesn't really kill *anyone*. not even the squishiest champs. Addition of Chilling smite doesn't help much. And going R early doesn't help: they just run to their allies or under turret. Does anyone have any personal experience of success to share? How to build? What to rely upon? Should I go auto attacks, like Tryndamere? Or should I build full tank and let others do main damage? Or change combo order, maybe? Or full lethality Rhaast? How to carry with Kayn as Red? {{champion:141}}
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