Someone really read this or not? So i try this guide for have feedback with a katarina build so please if you read this try it and tell the result in comment. So let's go!

Greater Mark of Magic Penetration and Greater Quintessence of Ability Power is for the early and late game to. For deal more magic dmg.
Greater Seal of Armor that is for tank the minions because after a Shunpo you can be focus by all minions.
Greater Glyph of Scaling Ability Power and this one is for get AP if you can't stack your Mejai's Soulstealer in your early.

A lot of other runs can be used to more AP more def like you wan't but i prefere this one for the reason i have give before.

I chose Mejai's Soulstealer at first item for have stack the most quickly possible.
then i chose Hextech Revolver for life stile and stay more longuer in lane.
the Sorcerer's Shoes bring the magic pen that i need for nullify resist magic of enemy carry
The Rylai's Crystal Scepter is for AP and for ability for slow after an aggressive Shunpo
Then the Rabadon's Deathcap givea lot of AP in this moment of the game you deal a lot of dmg and can stack mejai :D
If you are focus chose Zhonya's Hourglass (if you have get a lot of kill you are probably focus now :P) this active can save you wait the couldown of Shunpo and jump everywere you can get a kill of a low life enemy ,safe yourself, place a ulty after the all team use there cc ...
a MUST HAVE.
But if enemy team ignore you ... you can get a Void Staff for deal more damage but you are more squishy without Zhonya

For spell order before i have ulty i prefere max q for last hit minions.... And after level 6 max Shunpo for the cd decrease by the leveling of the spell and be more agressive after maz q and w for deal damage and up ulty when you can.

Flash
Why? because for escape an enemy you can had no minions in range and be in bad pos this flash is ONLY defensive if you wana be agressive use your Shunpo
Ignite because your w is better with your Shunpo because of bonus defense.... and agains't a soraka team Ignite can give you a kill and denied soraka ulty.

In team your are a killer :P the one who wait for enemy aoe and cc and Shunpo in enemy team and Death Lotus in and try to triple kill or more :P, you are defense for not being kill in ult or in lane but your are not a tank in Team fight. You can't jump everywhere: Shunpo deal a lot of dmg at end game but.. you have no escape if it is on cd so in TF don't shunpo on every enemy, except if they are low life and shunpo certainly kill them. In taht case... wit Voracity you have no cd on Shunpo and can shunpo kill shunpo kill ... ideal for end a TF if you was still alive ... after 3 kill your ulty has no cd to... so after a triple don't forget it.

So if you have read the whole guide.. sorry for my english i am very bad in this language but try to make it understable... I am a noob katarina and this guid is for me as for you i need training and know what think i have wrong,.. So test him send feed back and i change him and change my gameplay if it's better.