Not Updated For Current Season
Katarina is really fun to play, you can get a lot of kills with her, but you have to care, because she has low defense and can be focused down rather easily. So you have to practice her skills first to use them correctly in different situations. I will give you some hints how you can use them. Do not worry if it is not working all the time, just start and continue practicing!
This is my favourite rune combination for Katarina. Since Katarina does magic damage, magic penetration is well-suited for her, a little armor and magic resistance helps also a lot in terms of survivability and I use ability power quintessences to power up her skills.
I think it is good to have a little bit more defense at the beginning, so I put most points into that section and magic penetration helps in terms of increasing magic damage.
I think there is no optimal build, it always depends on your opponents, your playstyle and the whole situation you are in. This item builds should only give you a basic direction, whereof you should probably differ from depending on the factors mentioned above.
This item gives you a good boost to your HP, armor and also some life regeneration, which helps you in terms of survivability in the beginning.
Magic penetration means dealing more damage with your spells, so this shoes do a great job on Katarina.
Ionian Boots of Lucidity
If you prefer to use your skills more often, buy this boots, which give you a little cooldown reduction.
Buy this boots if the opponent team has a lot of stuns, slows and taunts. This item gives you better survivability in that case and additionally also a little magic resistance.
Boots of Swiftness
This boots are hardly needed, but in some cases, where the opponents have high movement speed you might want to buy this kind of boots.
Rylai's Crystal Scepter
This item really helps you in terms of survivability, because it adds HP and slows enemies on magic damage.
This item gives you a great boost to your ability power, so it improves your skills. Therefore it is quite powerful.
This item adds ability power, armor and the passive can be useful, when you are focused in team fights so they cannot attack you for some seconds while your team mates kill them and/or to wait for your skills to be available again.
Abyssal Scepter gives you ability power, magic resistance and a little magic resistance reduction.
This is also a good item which increases your ability power, movement speed and gives you a little magic resistance.
Buy this item, when your opponents stack magic resistance to have at least 40 percent magic penetration.
This item gives you magic resistance, armor and the passive revives you on death with a 5 minute cooldown. I think it is a pretty good item which gives you better survivability and the opponents might stop focusing you in team fights because of its passive.
Katarina's cooldowns are decreased when she kills someone or assists in killing.
Bouncing Blades is a quite good skill for harassing the enemies in laning phase.
This skill adds attack damage to her basic attacks and gives a bonus effect for the next skill used.
This skill is really great. You can use it for attacking to get next to an opponent and do damage but also to escape in dangerous situations. The combination of Shunpo and Death Lotus is quite good.
Katarina's ultimate damages nearby enemies over some seconds. It has a quite low cooldown, so you can use it frequently, also to kill single enemies. Rylai's Scepter really helps you to keep enemies nearby, so they can hardly run away when you start your ultimate, but it can be cancelled by stuns or some other spells, so be careful.
This spell is always good for escaping dangerous situations or chasing enemies, if the opponent has not many life left and you would not get him killed otherwise.
Second SpellThe second spell really depends on your team and your personal preference.
I really like Exhaust, because it is universally applicable. You can use it to attack in 1v1, in team fights to take out their main damage dealers, but also to escape from heroes chasing you.
Ghost or maybe also Ignite are other good options.
Katarina is really cool. Her skills are applicable in quite different situations. She does lots of damage and you can get a lot of kills by playing her. If you do not believe me, try her out!