Katarina is a very good crowd control champion if you play her right. So, Katarina has to be one of the most lethal and best characters when played with the correct team. Any stuns or slows paired with Katarina is a lethal combination.

Ok now consider AP, with this build, you have 580 AP, Now concider you also have 60% spell vamp Oh boy. Now, consider your ult is 30% of your AP, and 55% of your attack damage 288*0.55= 158 + 580*0.3 = 332.4 damage per dagger.. and you steal 60% of that to your own hp, so 332.4*0.6= 199.44 life stolen per dagger. Wow go figure, and guess what, your ultimate, and your shunpo, and your dagger slows. This is because you got that handy rylai's sceptre. Now shunpo damage. 210 base at lvl 5. plus 75% of the 580.. well thats, 435 + 210. So your shunpo is hitting 645, and spell vamp 60% = 387 life to you self per dagger. woah, you can probably 2-3 hit a squishy character with that, with a 35% slow for 2.5 seconds?? HOLY ****.

Boots:
I would always prefer sorcerer's shoes, but if you are against no magic resist you can also take ionian boots of lucidity.

Items:
Rylai's scepter, is the ultimate item for katarina because it slows your enemies when you attack with magic damage so when you are using death lotus all hit enemies are slowed. With 3 hextech revolver, you got 45% spell vamp, so again when u use your ultimate you gain alot of life. Rabadon's deathcap, is just for the nice ability power bonus. If you go late game and get full items, then the hextech gunblades are very usefull because your ultimate also scale with attack damage.

Earlygame:
Earlygame is all about last hitting minions, avoid to die and maybe get firstblood. Bouncing blade and shunpo is very good for last hitting, when you hit level 6 and get death lotus, if you have an allied with a taunt or stun like rammus, galio or sion go for the kill start with activate killer instinscts shunpo to your enemy let your allied stun or taunt use your ultimate, while the target is stun or taunted, if there still not dead you have bouncing blade to last hit, or else your shunpo cooldown should be down be the time.

Midgame:
Midgame is all about getting kills, pushing towers and remember still last hit minions. Remember you shall never start a teamfight, when the fight is startet shunpo in use bouning blade then ultimate if you have full hp continue is low hp flash out.

Lategame:
Lategame is all about winning the way to victory is to destroy towers, and the way to destroy towers is kill your enemies first and the way to kill you enemies is katarina. Your team have started a team fight katarina arive, you use: E, W, Q, R, and all is dead.

Flash:
Flash is very good to gangk and flee. If your in a team fight and you shunpo in the middle off all people, use ult flash away.

Ignite:
Ignite is very good to get first blood, remember first use your ignite when your target is fleeing. Remember that ignite reduces healing done too your target so if you are against dr. mundo, swain or vlademir etc. use it on them so their healing abillities are reduced.

Bouncing blade:
Is a very good ability to last hit minions. Remember bouncing blade reduce healing(if you activate Killer instinscts first) at enemies, so start a team fight using bouncing blade, when it is full upgraded it can hit all five enemy champions.

Killer instinscts:
Passive: Base damage increased.
Active: Makes your bouncing blade damage full damage at all targets and reduce healing very good to start a team fight with. when you shunpo all damage to your are decreased so if you start a team fight use that.

Shunpo:
Shunpo teleports you to your target location, damage well too your enemy if you level up easy in the game to can easy take half life on weak champions like twitch. It is very important to remember that you can shunpo on allies and allied minions to escape. Shunpo is also very good to last hit minions, just remember you land behind your target when shunpo it.

Death lotus:
You start throwing knifes around you and it scales both attack damage and ability power, if youhave rylai's scepter, it slows your target(s). Use death lotus after shunpo in the middle of all.

My rune page is very overall, u can use it for almost every champion.

Quintessence:
Health, Flat health is very good in the beginning makes you less killable, and gives you the control in the start if you have the opper hand, you have it more easy to push and get firstblood.

Seals:
Dogde, Flat dogde is very good in the whole game special in the beginning, if you have good dogde you are less killable.

Glyphs:
Cooldown reduction, flat or per level is your own choice i prefer flat because there is not the big diffrence between it. Katarina's abillities doesn't cost anything, neither mana, energy or health. So you can just spam use your abilities and if you have less cooldowns you can use them more.

Marks:
Magic penetration, flat magic penetration your abilities damage more, all champion have base magic resestance so you can use it against all champions.