Katarina is a very strong pick and people feel the strength of the pick as they will usually, if they can, try and base their team composition on trying to shut her down. This is a very hard thing to do, but it is also very hard to get her rolling if you pressured very hard in the early game.

Katarina can be pressured very hard by an enemy Mid laner that is hard to gank and is able to push your lane very hard. Heimerdinger is a very good example of such a champion as it is really hard for a Katarina to do anything meaningful to him without losing a substantial amount of health in the process since she only has one gap closer, low damage on her Q at early levels, and really no way to out-push a Heimerdinger.

Kat can also be pressured by ganks; is their jungler an early-game play maker? like Lee Sin or Elise where they will be constantly ganking lanes and trying to create leads? Are they aa farming jungler that will rush 6, complete their smite tool, and then gank a lane and take that turret? Are they a tanky jungler with a hard-engage ability or a gap closer that can catch her out? Or are they a jungler that is so tanky that they can tank turret and go for a dive? These are important things to know since they determine how aggressive Kat can play at certain points of the laning phase and it will also determine where they are going to need to put wards.

Katarina is also pressured by other Assassins since they have a way to either negate the damage from her ultimate, they can out-maneuver her, or their ultimates are simply more potent than hers without any items. Playing against a Zed is really hard for her since it's sometimes impossible to kill him while he can one-shot you, or just get away from your ult's range unscathed. Katarina really thrives against immoble mages since the only way that they can negate her damage is with CC, and that requires a high reaction time.

If you think that you can farm out the lane and get Luden's without dying, then you can destroy the enemy Mid, or you can just roam to go destroy other lanes--it doesn't really matter.

Kat isn't very reliant on levels and can go for a very early all-in on a champion and win with flying colors but this has to be done either at level or once you grab your first big item.

-Runes-
These Runes are hybrid runes, and for good reason.

9 Marks of AD
5 Seals of Armor and 4 of MR
6 Glyphs of AP and 3 of MR
and 3 Quints of AP

Why AD Marks?
Katarina has two abilities that scale with AD, one of which you use to your advantage at very early levels. Kat's W scales better with AD than with AP, and because of these runes you will have much or AD than AP until you get the Large Rod. This also makes it easier to farm because you can mark something with your Q, and it will do more damage early in the game than it would when you scale with items--this paired with her W will make for quite easy poking.

Mixed Seals?
I had to sacrifice some armor for MR because most of the champions that go mid are mages and AP assassins, so you are gonna need some MR. The reason why I had to mix them is because I also mixed the Glyphs by exchanging some MR for AP, which I think is actually pretty effective since you will get a moderate amount of AD and AP in the beginning of the game. This allows you to play more aggressive, but remain aware of what you've exchanged for damage.

Glyphs
AP Glyphs are very effective since they give you the most bang for your buck giving you a whopping 1.2 AP a piece. This rune gives the highest damage stat of any other, not including Quints, in the entire game. Since spells require more AP to do more damage, since MR scales differently than Armor, and spell damage being generally more potent than physical damage, lots of AP is valuable.

-Masteries-
The mastery trees are both the same since they both try to serve really only one purpose--to survive the early game. These masteries allow you to trade and farm quite well. You need at least the 21 points into offense because if you didn't have that, then you wouldn't be doing any damage until late game and really wouldn't scale that well. Try playing a game without the Havoc mastery on and you'll see the difference in damage immediately--it's pretty significant.

Havoc and Devastating Strikes are completely necessary masteries to have points in on champions that are meant to do damage because otherwise they wouldn't really be doing much. The damage you do without these masteries only pokes at their health bar instead of really chunking any amount off.

I would really like to point out the Blade Weaving and Arcane Blade masteries being used here and how it helps Katarina with her early game weaknesses. Trading can sometimes be difficult since you have to get close to your enemies to do it effectively, and you only have one gap closer. The fact that you only have one gap closer means that you don't have anything to create distance between you and the enemy mid laner. What you really need to do is pay attention, very close attention, to their cooldowns and determine if whether or not it's safe to go in. When you go in, the damage you do with basic attacks will be enhanced due to these two masteries.

Blade Weaving is basically the same as Riven's passive since it stacks 3 times with the usage of spells but instead of being able to expend it 3 times, it stacks up to 3 times to do 3 times more damage. When going in on an enemy, or more specifically, when trading, Katarina quickly uses 3 abilities. By the times she's used all three of these abilities she will be right next to her target, and can easily auto attack them and expend those Blade Weaving charges.

Arcane Blade is a really strong mastery that scales really well with lots of AP. This mastery basically says that once you get Deathcap, your basic attack damage will have a noticable increase--this increase in damage makes it easier to push turrets. You can split push turrets as a Katarina later in the game if you get this mastery. Arcane Blade also works very well with Blade Weaving and increases the damage you do during a trade very slightly--enough to make a noticeable difference though.

I decided to put points into the defensive masteries opposed to utility since Katarina isn't a ranged champion and needs to get close to her enemies to do significant damage. Being able to take as much damage as possible is very important when surviving a potentially difficult lane. You could get utility masteries if you got a Doran's item, but you won't be so defense is definitely the way to go.

-Kat's Q-
Bouncing Blades is the strongest ability that Katarina's kit possesses because not only does it have a high initial damage, it's reactivation causes it to do much more bonus magic damage. Bouncing Blades makes everything that Katarina does hurt like hell even if her other skills or her auto attacks aren't really that strong, the reactivation damage proc scales very well. The total potential damage that comes from this skill is what allows Katarina to dish out lots of damage in less than a second.

Bouncing Blades increases Kat's trading ability greatly because of it's double proc. She doesn't even need to proc it the second time to be able to trade effectively, but if she can, it's quite a large amount of damage that's expended so quickly that it's sometimes quite hard to respond to. This ability also accounts for all of her wave clear and ability to farm with ease.

-Sinister Steel-
This ability, actually, scales better with AD than it does with AP and the reason why stacking AP is stronger is simply because of the high numbers of AP that Katarina can reach with items. Getting a Long Sword at the start of the game works quite well because AD is more effective on Katarina, especially early on in the game, than AP because she is able to reach 100 AD very quickly.

Sinister Steel scales with 60% AD and 25% AP and the damage that it does, when paired with Bouncing Blades, especially in the early game, is quite high. It's actually higher than just going full AP at the start of the game, and is overall better for farming and trading. Katarina can use her abilities and her basic attacks to do damage opposed to just going full AP and relying on her spells from level 1-4.

Maxing Sinister Steel second is more advantageous in most cases since it increases the amount of AOE damage that Kat is able to do and ultimately increasing her AP scaling. She will be a lot less mobile, but it's worth it in most cases.

-Shunpo-
This spell is what Kat needs. The cooldown is quite long if not skilled much, and it accounts for a large chunk of her damage since it's the spell that she uses to engage. The damage being done is very high since this spell is being used right after Bouncing Blades. Her W has the potential to do lots of damage, but it's a lot less valuable than Shunpo.

Shunpo's cooldown, at max rank, is 6 seconds and does 160 (+25% of AP) magic damage to a target. This ability also decreases all sources of incoming damage by 15%, making it a really great spell to use as often as possible. Shunpo also happens to be the ability that allows Katarina to Pentakill by jumping around fights and dealing massive damage in team fights if she is able to get her resets off.

It is actually better to max this skill last in most cases except for the ones where Kat has created a very large lead for herself.

-Death Lotus-
Death Lotus, Kat's ultimate, deals massive damage to up to three nearby targets in an AOE. This ability is a channeled ability that takes away 100% of Kat's mobility, canceling the ability entirely and putting it on cooldown is she moves or uses another ability prematurely. She can use items like Gunblade that have a single-target effect or the summoner spells ignite and smite during Death Lotus' channeling.

Death Lotus scales very very well and will do well over 2000 magic damage at full build and close to that number with 3 items and at level 16. It's base damage is very high being 350/550/750 at levels 6/11/16 respectively and scale insanely well with a 2.5 (250%) AP scaling and 3.75 (375%) AD scaling. I know that Kat's ultimate will apply effects from abilities on the people in it's AOE, but I have yet to find out if it applies on-hit effects as well such as lifesteal or critical strike chance.

-Gunblade Kat-
This build gives Katarina spellvamp, which works almost too well with her ultimate, makes her a very potent team fighter and makes her impossible to kill without CC. If she isn't one-shotted, she will just vamp the health that she loses by using her ultimate on at least 2 people. 20% of over 2000, which is how much damage her ultimate will do later in the game, will heal her fully in a matter of seconds.

[*] Hextech Gunblade is the item that allows Kat to heal up for such a great amount, and scales her up just a bit better than usual due to it's hybrid nature. Since Kat scales better with AD, the 40 that Gunblade gives her is very beneficial to her growth throughout the course of her build. Gunblade's active ability is very good for chasing enemies and is also something that scales with her AP, actually doing quite a bit of damage once she builds items; Gunblade's active does 150 (+40% of AP) magic damage to the targeted enemy. At full build, that active will scale up to be about 500 since Kat will have about 800 AP.

[*] Luden's Echo is the item that Katarina needs to get first since it greatly increases the amount of burst damage that she is able to do. She will have over 100 AP at this point, which is pretty high for having only one item, and will be doing about 300 damage a target every time Luden's is at 100 charges. The movement speed buff that Luden's Echo has makes Katarina very very fast and will be able to roam very quickly. This item is literally made for Katarina since it gives her everything she needs at the early stages of the game to make her very strong after building this item. Luden's allows Kat to build items that don't have lots of AP because it makes the most of what AP she has, which, after building it, isn't really that much but remains the most effective and efficient item that Kat can get.

[*] Liandry's Torment is the item that Kat will pretty much always need to get second because it greatly increases the damage that she is able to do because of it's % current health DOT. This item is potent, but not just because of it's fiery passive, but because of the Magic Penetration and pretty high AP stats that it gives you being quite an effective item at all points in the game. This item is really about maximizing DPS, and really does so quite well since it makes Kat do considerable damage to pretty much everyone, even tanks, due to it's passive that scales off of health. The magic penetration stats will be very strong against squishier targets since their resistances will be low pretty much at all points of the game, unless they get a GA, Banshees, or Visage--even if they do, the damage Kat does will still be quite high because of this item.

[*] Rylai's Crystal Scepter is an item to get after Liandry's because it's supposed to make it stronger by making it twice as effective. When a target's movement is impaired, the damage that Liandry's passive does is doubled--that's 12% of a target's current health just gone from using an ability on them. This item is basically just used in unison with Liandry's; their passives rely on each other to maximize effectiveness, and their HP stats make Kat quite a bit tankier. She doesn't have MR or Armor with this build, but HP does reduce her squishiness greatly. These two items give Kat a total of 700 HP, 2621 total HP, without runes and masteries, at level 18. With the runes and masteries of this build, Kat will have a total of 2737 HP at level 18; she will be quite tanky for a Katarina.

[*] Rabadon's Deathcap is the last item that Kat will ever get since it's very effective later on in the game. With Luden's, Liandry's Gunblade, and Rylai's, Kat will have 400-500 AP. After Deathcap, she will have 700-800 AP, which is close to double what she had prior. Everything that she does at this point will hurt ALOT, and will be one-shotting squishy targets with two abilities, only really using her ultimate in large team fights.

-Holy Damage Kat-
This build is pretty much the same as the other, except instead of having Gunblade, she has Zhonya's. The abilities that scale with AD will do less damage, but it's really not a big deal considering what Zhonya's allows Kat to do.

[*] Luden's Echo is Kat's first item and will increase her burst by a great amount early game, and a deadly amount in the later parts of the game. The Luden's passive scales with 20% of Kat's AP, so she will definitely have no shortage of burst. The movement speed this item gives is amazing as well and is quite valuable, especially at the time of purchase since it will make her extremely fast.

[*] Liandry's Torment and Rylai's Crystal Scepter are the next two items on Kat's purchase order since they work extremely well together and give her lots of HP. Liandry's is to always be purchased first since it is more than 100% cost effective and increases Kat's damage by a large amount. Rylai's comes after to further scale her AP, HP, and Liandry's passive making it do double the damage it would normally.

[*] Zhonya's Hourglass, at time of purchase, makes Kat very hard to deal with in team fights and will make her quite a bit tankier due to the Armor stats it gives her. The 50 Armor from Zhonya's and the 700 Health from Liandry's and Rylai's will make for one scary Kat. The Zhonya's active can be used to avoid long cooldown ultimates, fatal CC abilities, and also very fatal and high damage ultimates with the click of a button. Most people won't have the time to wait around for 2 and 1/2 seconds for the Zhonya's active to dissipate meaning this item will allow Kat to initiate fights.

[*] Rabadon's Deathcap is always to come last since it is really the icing on the cake and will scale Katarina's AP through the roof, even more so with this build since it will come to a total of about 800-900 AP. Her ultimate and her W won't scale as well as they would with Gunblade, but her burst damage will increase due to the Luden's passive. This build is certainly allows her to be more aggressive in fights.

In this chapter I'm going to breifly explain how to position in fights, when to go in, and what each build will allow you to do/restrict you from.

Katarina is an assassin, a very vulnerable one at that, and is very reliant on her team to go in first. This is because of the fact that her ultimate is very easy to cancel and she is often very easy to kill since her Shunpo ability is her only gap closer. When Kat goes into a team fight, she always goes all in since she doesn't have a single disengage ability aside from Shunpo, which also happens to be the same ability she uses to engage. When CC'd, she will almost always die, until she grabs some items, unless her team is able to counter engage with a large CC ultimate or something like an Amumu or Malphite.

Katarina will need to sit off to the side of team fights, waiting for the perfect time to go in. This perfect time would be when the enemy team is grouped up and have used their CC abilities and ultimates. Due to Kat's large amount of AOE damage, she will be able to carry team fights by just going in after there is nothing left to stop her ultimate. The focus will immediately shift to Kat, and will allow her team to counter-initiate into the fight and assist Kat with getting her full ultimate's channel off.

[*] Gunblade Kat will allow you take more damage in team fights, chase any stragglers with ease, and become quite hard to kill due to the large amount of spell vamp she has. In a fight, if she is low and has survived a long cooldown CC ability, she can re-engage after her team goes in and end up with near full health after using her ultimate. She heals for so much health so quickly that really the only way to kill her is to cancel her ultimate. In every build, Kat will be building Rylai's Crystal Scepter, which will help her get her ultimate off on multiple targets since they will be slowed a lot by it.

[*] Zhonya's Kat will allow her to be the primary engage for her team. She can go in without using her ultimate and use Zhonya's immediately. While she is untargetable, her team can follow up, which will make it hard for the enemy team to focus her. Since they won't be able to sit around and wait for Zhonya's active to dissipate, they will pretty much be forced to use their ultimates and CC abilities on your team.

With Katarina, don't ever go in first. It's not worth it. She can easily get Pentakills by just waiting by for that perfect moment to engage instead of being super greedy and becoming the initiation.

Don't misunderstand though. Kat snowballs EXTREMELY hard, and creates such insane leads by playing super aggressively in fights. If she is ahead, and going in first would be very successful (in a dive, a small skirmish, etc.) then it would be very advantageous to do so. Just make sure that you understand the situation before you throw Kat into a fight.