This Build is meant for Katarina to do TONS of damage and yet keeping in mind to playing defensively. This is my first guide so I did not really put in a lot of details in this guide but this build is extremely effective against most AP champs.

In this build, Katarina gets tons of AP, that it powers up her extremely. With the health from Liandra's Torment and Rylai's Crystal Sceptor, she can sustain in lanes a lot longer than most AP mid champs. This build works both early game and late game, basically making her an all game power champ. The slightest advantage in AP or health will effect the outcome of a fight so this build focuses on both of those, the rest is up to how you play Katarina.

Voracity (Passive): Champion kills or assists reduce Katarina's cooldowns by 15 seconds.

*Note: Really useful, all your basic skills cool downs are less than 15 seconds and your ult already has a short cool down.

Bouncing Blades (Q): Katarina throws a dagger that bounces from enemy to enemy, dealing magic damage and marking them. Her next spell or basic attack against a marked target will consume the mark and deal additional magic damage.
Throws a dagger dealing 60/85/110/135/160 (+45% Ability Power) magic damage. The dagger bounces to the 4 closest enemies dealing 10% less damage with each bounce.Enemies hit are marked for 4 seconds. Katarina's basic attacks or spells will consume the mark dealing 15/30/45/60/75 (+15% Ability Power) additional magic damage.

*Note: Main Poke Skill

Sinister Steel (W): Katarina whirls her daggers around her, dealing magic damage to all enemies in the area. If she hits an enemy Champion, Katarina gains a burst of speed for a short duration.
Whirls daggers in a circle dealing 40/75/110/145/180 (+60% bonus Attack Damage) (+25% Ability Power) magic damage. If she hits an enemy Champion, Katarina gains 15/20/25/30/35% movement speed for 1 second.

*Note: The movement speed boost is nice. The damage is decent but low at early levels.

Shunpo (E): Katarina instantly moves to her target's location and takes reduced damage from enemies for a few seconds. If the target is an enemy, she deals damage.
Moves to a target's location. Deals 60/85/110/135/160 (+40% Ability Power) magic damage if the target is an enemy.After using Shunpo, Katarina takes 15% reduced damage for 1.5 seconds.

*Note: Gapcloser/Escape Skill. Great Synergy with Sinister Steel. I think it was based off of Bleach :D

Death Lotus (r): Katarina becomes a flurry of blades, throwing daggers with unrivaled speed at up to three nearby Champions. Daggers deal magic damage and reduce incoming healing on targets hit.
Becomes a flurry of blades, throwing daggers at the closest 3 enemy Champions dealing 400/575/750 (+375% bonus Attack Damage) (+250% Ability Power) magic damage over 2.5 seconds.Daggers apply Grievous Wounds, reducing incoming healing by 50% for 3 seconds.

*Note: Main Damage Skill. Deals TONS OF DAMAGE. Short cool down. Easy to interrupt though. Position yourself wisely.

Core Build should give you about 250 AP plus (The passive of Rabbadon's Deathcap) + 75 = 325 AP, and if you add Will of the Ancients (has the least amount of AP given in the situational section) you would get a base of 300 AP plus 112.5 AP = YOUR LEAST AMOUNT OF AP FROM A COMPLETE BUILD: 488 AP then if you add what the Runes and Masteries give you, you would at least get 500 AP with Kat. You would get 800 bonus health and 138 from masteries which would get you 938 and with your base, your total health would be 2827.

Pro:
1)
Gives you tons of burst from your ult and makes you less squishy. Every game I won as Katarina I would always have at least a triple kill.
2)
Manaless.
3)
Wardjunping (not really a jump though)
4)
Strong Teamfight influence if played correctly
Cons:
1)
Kat's ult can get disrupted, leaving you to use your basic skills, which does not have consistent burst (This is not because of my build, Katarina is just like that). This build will not work that well if you are behind.
2)
Easy to lockdown
3) A possible tough laning phase pre 6

Rylai's Crystal Sceptor: Really useful item for Katarina. Helps you stay in lane longer and the ap is pretty decent. Plus this item is gold efficient. This item is really useful Early-Mid Game. In late game, this item is not as useful as before but it can help annoy the enemy team with your bouncing blades.
Rabbadon's Deathcap: No need to explain. Useful for ALL Apc.
Liandra's Torment: Magic Penetration is nice and helps you stay in lane.

Katarina's standard combo is q, e, w. You use your q for poke and then e to pop your q mark. Your w is a given every time you use your e on an enemy champion. The full combo including ult is q, e, w, and r.

The one thing you never do as Katarina is initiate a team fight. It would be the most stupidest thing to do. You are the main damage source for your team and you don't want to die. Your ult is the most useful thing in a team fight since it fires at 3 champions while dealing tons of damage. It instantly melts your enemies. You should try to use your e to your advantage because if you e into a group of champions, and they are not focusing you, you can ult and it would be hard for the enemy team to switch focus to you since you are in the middle of lots of champions. Katarina should be the last person in your team to actually fight in a fight. It is the most effective way.

Farming with Katarina is easy. You can use your basic skills to farm as long as you aren't vulnerable to a gank. Your q is your most useful farming tool and it helps poke. This is one of the pros for Katarina… Manaless… and decent cool downs.

Just to let you readers know, or if there are even any readers, this is my first guide and I know its pretty bad since I did this in 5 minutes. Anyhow, I would like to know what you think about this guide.