Although she has been nerfed in the past, Katarina still behing a great choice at time to do massive damage as main AP carry of your team. For all that you will need:
-Katarina
-Common Sense (don't focus the op thank!!!!!)
-A decent team comp to keep your Death Lotus active as long as possible in team fights

Katarina Pros:
-No MANA!!!
-Low cooldowns
-AoE ultimate
-If the other team has no CC's to disable your Death Lotus you win

Katarina Cons:
-Squishy ( Rylai's Crystal Scepter is usefull not only for the AP + Slow. Gives HP wich is good to survive and that is why i put Hextech Gunblade)
-Not a good farmer at early game
-You will be the first focus on every team fight
-You need good partners to keep you alive (specialy when you activate Death Lotus)

Common runes for an AP carry. Greater Mark of Magic Penetration gives you a nice MP (magic penetration). Greater Seal of Health and Greater Seal of Magic Resist gives you a nice survavilty for early game. Greater Glyph of Cooldown Reduction reduces the cooldowns and Greater Glyph of Ability Power gives flat ap. All this plus the Greater Quintessence of Ability Power will give you a nice ap damage on early game.

A classic 9/21/0 to get MP of Archaic Knowledge , Survavilty and a bit of AP that Ardor gives. Not too much to say... just this.

1
Ionian Boots of Lucidity:
I take this boots due his Cooldown reduction. I found them more usefull because they work preety well with Voracity. Another important thing about this boots consist on the utility of the blue buff. I preffer to leave this buff to a mana dependant characther. So... leave this buff to your Nocturne, Amumu or Janna. They will apreciate the mana regen to stay in the battlefield more time.

Options:
Sorcerer's Shoes whorks well to make you more scary. They give's you MP wich is great to make more damage on a enemy who is stacking magic resistance.
Mercury's Treads
Not too common, take this if you are laning against an AP caster. Althoug they can reduce the CC time against you, they are just usefull on 1v1. in team fights u are the first focus and this will be not enough.

2
Hextech Gunblade
This is an awsome item for Katarina. Gives a decent AP at early game plus spell vamp. If you are low of HP pop a Bouncing Blade into a wave of creeps, sunpo and BAM! you get some HP recovered. His utlity while you are casting Death Lotus is great too.
I recomend to finish this item after you get Rabbadon's deathcap. why?? because is an expensive item and you need to maximize your ap quickly. Hextech Revolver is what you need of this, lets say it's the "why" i get this item.

Alternative: Will of the Ancients
Is a situational item which is great if your team got another caster like Veigar, Ryze, Malzahar or LeBlanc. Believe me: this can be a key item in the game.

3
Mejai's Soulstealer
Snowball item and like all the snowballs it can be awsome or he can be a wrong choice. They can be the cheapest item or the more expensives. A good signal to take the decition are the first ten minutes of the game. Analize your opponents, the champions. Are they playing agressive? how much disables they got?. Like a kind of parameter i take the very early game: if you got 2-3 kills before the first recall i MUST take Mejai's Soulstealer. If you are getting pushed, harrased, or your lane partner sucks bad, forget it. go for rylai's and save the slot for another Rabadon's Deathcap or a Banshee's Veil. It has no sense to had a snowball if you can't grow at early game.

4
Rylai's Crystal Scepter
Great item cause he gives to katarina one thing that she doesn't have and its one of the most important things on LOL... it calls CC (crowd control). Rylai's not only gives 80 AP and 500 hp... no no. After this you will keep your enemies in the range of your Death Lotus due the slow that this little bastard gives. If you don't get the kill means that your opponent has used his Flash or Ghost at exchange of one ability that has a ridicolous Cooldown and decreases if you get kills or asistances due Voracity

5
Rabadon's Deathcap
Your MUST. No AP caster on LOL plays without him. 155 AP and his more important atribute: 30% of your AP increased. Make the count: 160 of your Mejai's Soulstealer, 80 of your Rylai's Crystal Scepter, 60 of Hextech Gunblade, 70 of Void Staff and of course the 155 AP that the same Rabadon's gives you. If you rush your mejai's you will get around 800 of ap. Remember that you have Ardor in the masteries wich gives 4% of increased ap.

6
Void Staff
Situational item: get it if your rivals are stacking Magic Resist to blow up his armor (make that Force of Nature or Banshee's Veil a piece of ****)

Options: At exception of AD items any option it's correct if you read the game properly. But a help:
Banshee's Veil or Quicksilver Sash if their caster is afecting your game
another Rabadon's Deathcap if the enemy is squishy and it's not stacking MR (you may have a pentakill if you make your Preparation / Shunpo / Death Lotus combo in the right time)
Defensive items wich are not MR. Use them if the enemy have many ad's wich can interrupt your ultimate or killing you instantly. (i know this next can sound ridicullous but...) A reasonable choice is Thornmail but if you need to get this item or Warmog's Armor means that someone in your team it's not doing his job well.

Bouncing Blade
I maximize this ability first due his farming an harrass potential. It's a usefull ability to reduce the Hp of your oponents until you decide that it's time to get the kill. It's great against turret huggers: just wait until the enemy creeps come, Preparation and Bouncing Blade and that coward will get the punch that he deserves.
Katarina is not the best farmer at early game, that's why i maximize this ability first. Of course that you can maximize Shunpo to get a more "nuker" ability.

Preparation
I only spend one point at level 4 to reduce the hp regen once i pop Bouncing Blade but don't underestimate this ability. His utility will be increased at late game due the combo with Shunpo wich will give you an incoming damage reduction to allow you cast your ultimate in teamfights for 3 seconds.

Shunpo
One of the greatest abilities in League of Legends. Completly versatile to enter/escape from teamfights, chase enemies and more. This ability is too the reason why i ALWAYS play AP Katarina. Don't misunderstand. AD build can be usefull but honestly i've never played in this way with kat. I'm trying to make a guide very short and concrete but i got to say that Shunpo deserves a entire chapter just to explain all his potential. I gonna put some examples:
- 75% of your ability power scales with the 240 that sunpo gives normaly. Means that at late game (if things are going good) you will do around 750/800 magic damage just with sunpo.
- If you are in troubles you can Shunpo into your creep wave or an ally and scape of a bad situation. You also can enter into a teamfight over the main carry and activate your ultimate.
IMPORTANT:
In teamfights activate your W ( Preparation) before Sunpo. You will get 35% reduced damage from all sources in the next 3 seconds. This is a lot of time, considering that the teamfights resolve themselves in shorts periods of time (at 20 seconds shoulda be decided who is the winner).

Death Lotus
This is one of the most dangerous ultimates in the game and the reason of why you will be focused in mostly of times. Requires good judgment which means one important and valious thing Katarina NEVER NEVER NEVER INITIATE. Believe me: many players do this and they die instantly, even with the W-E combo before R.
Death Lotus is an AoE ultimate, scales with AP and AD damage, consist of a decent area of impact, is viable to combine with other champs abilities. ( Curse of the Sad Mummy or any snare or AOE Stuns like Event Horizon or Solar Flare are the best choices).
Use Death Lotus when the enemy cast his CC against other teamates. That should garantize kills to you and your team: the obvious result you win the teamfight, u get stacks, u get gold and the chance to push to TADA!!!: WIN.
The great Con of this ability: you are the first focus and that is why you need to engage a few seconds before that your team in the 5v5 encounters.

MAIN SPELLS:

Flash: altough you got Shunpo you need flash to scape in the jungle or that situations when you don't have any creep or champion or NOTHING to point. Is the most used spell in League of Legends, great to scape, great to chase, great to engage. Why not???

Exhaust: this is a good spell, specially to keep your enemies on the range of your ultimate in early game when you don't have Rylai's yet or against hard dps who can take you down in matter of seconds. Is good but they are other choices.

Plausible Spells:

Ignite: always usefull to get first bloods, or those ****ers who run away with 10 hp.

Teleport: this spell is more usefull that i've could imagine ever. Mainly is used to keep you in the lane. But i find it good to go to a skirmish if you have a ward near. With a Teemo on your team, this spell is awsome (i'm thinking in the shrooms).

Ghost: same function that Flash, many people see this spell better than Flash. I don't but fell free to pick.

Cleanse good against many cc's, but your team must take care of you and avoid your death.

Clairvoyance great to prevent ganks. Anyway you are not who must take it, but it can save your life

Spells to NOT TAKE:

Heal: you got spell vamp and this spell is usefull at early game when it suppose that you are not dying but killing.

Rally: you are not an ad

Fortify: leave this to your tank, you are the carry

Clarity: lemme see.... why not? Oh i know!!! KATA DOESN'T HAVE MANA!!!!!!!!!!!!

Smite: You are not a jungler.