Katharina is one of the champions you should not feed while playing v her. She has one of the strongest combo's ever and with teh right items her damage output is enormous.

To start off with we shall look at her skills seperately.

Bouncing Blade: Katarina throws a dagger which bounces off to multiple enemies. This deals less damage each subsequent hit.

Killer instincts: If katarina activates this skill she has bonus damage, not only bonus damage is given though. When Katarina activates Bouncing Blades or Shunpo, this has a special effect.

Shunpo: Katarina jumps towards an enemy dealing damage

As you can see i prefer to first upgrade bouncing blade because of the harassing capabilities, taking a level in Killer Instincts at level 4. Kata is one of the best solo champions just because of these harassing capabilities + she doesn't use mana nor energy.

I play Kata as a full ap champ, but it is possible to play a kata ad too. This hybrid katarina is very compatible because the percentage of ad is higher than the percentage of ap