With all masteries on Offense Katarina will got crazy dmg at start of the game with melle dmg and AP close to 40 and increased about 3 per lvl with shunpo you can reduce their life and making easy the first kills also the healing potions will help you to stay more time at lane and the boots will help you to escape after using shunpo.

Buying first item Hextech and after this make Shen this item gona increase your burst dmg much so you can kill them more easy. When you make the Hextexh revolver you gona get 1 skill with 300dmg and slow this will give you many kills. Making Will of the ancients next item will increase your survivability when using your ulti and also helpyour AP teamates. After those items you gona get close to 300AP and its the time to make Lich Bane with this item you gona make CRAZY dmg at everyone you attack with every time using skills you gona heal yourself close to 200hp ( Shunpo ) You gona get everything you need with those items Slow by ( Hextech Revolver ) Spell Vamp , Crazy Burst dmg with Shunpo+LichBane After those items you can make another Hextech just to get 60% spell vamp or you can make Rabadon/Rylai's/Void Staff Buy whatever of those items you think gona help you on late game!