I'd like to start off by saying that Katarina gets a bad rep and gets bashed for being so easy to get kills with and how over powered she is but in my opinion playing Katarina requires ridiculously fast fingers and equally quick decision making and knowledge of every matchup and damage in every stage of the game. Katarina is not as easy as she is made out to be. Period.

The rune setup I use on Katarina mid is one that is just made to be flexible and do well against any sort of enemy laner.

Greater Quintessence of Ability Power
Ap quints are pretty much the only way to go on Katarina because they increase your damage alot in the early game and doing damage is what your made to do. Move speed quints could also be applicable.

Greater Seal of Armor
I like to take armor seals for a bit of resists from an ad laner that i may be agaist like Zed or Riven. It also reduces creep damage a bit which helps more than you twould think in some riskier plays. scaling or flat health seals could also be applicable.

Greater Glyph of Magic Resist
Since in the mid lane its most likely that the enemy laner is going to be doing magic damage its a pretty good idea to give yourself some better resists for them.

Greater Mark of Magic Penetration
I like to take magic pen marks due to the fact that its pretty likely that the enemy laners rune page includes MR as well, the Mr marks provide you with a solution to this.

STARTING ITEMS:
I in almost all cases start Boots and 4 Health Potion but Cloth Armor and 5 Health Potion can be better against matchups with alot of ad based harass like Jayce.

CORE ITEMS:
With the removal of Deathfire Grasp the standard first item for Katarina against AP I think should be Abyssal Mask, however on your first back if you can afford a Needlessly Large Rod then you should and then go into the abyssal, if your against an AD Laner then an early seeker's armgaurd can be great, this will be built into a [[zhonya's hourglass later, A death cap should be your second or third item pretty much every game for the reason that it gives you more AP then any other item in the game, but because of the passive it should be built second or third.
So basically in most games your first items should be:
Abyssal Mask
Rabadon's Deathcap
Zhonya's Hourglass
Zhonyas and abyssal can be swapped against AD.

DEFENSIVE ITEMS:
If you choose to take a defensive item as your last item its a pretty simple choice, gaurdian angel for a balanced enemy team comp Randuin's Omen for a AD heavy enemy team comp and Banshee's Veil for AP Heavy enemy team comps

CONTROVERSIAL ITEMS:
Twin Shadows can be a god item to replace boots in the way late game. Will of the Ancients I personally believe that spell vamp is often overlooked on Katarina due to the fact that almost all of her abilities are aoe so the effects are only 1/3 but i still think that WOA can be great on Katarina because it allows you to stay healthier while channeling your ultimate. Rylai's Crystal Scepter Ryali's is often overlooked on Katarina for a similar reason to WOA and that being that the slow effects of Rylais are reduced because of Katarina's AOE, I still find myself building this item every know and then however because it gives a very big chunk of ap as well as a little bit of survivability from the health but it also forces people to need to sit in my ultimate for longer, Haunting Guise is an item that many people build on Katarina and swear by but I personally don't think the item has a place in most Katarina builds. Its still a strong item that gives a lot of good stats but there are more, better items to build. Hextech Gunblade Gunblade should never be built on Katarina. Period.

OTHER ITEMS:
As far as boots for Katarina i think that Sorcerer's Shoes are the best option because it allows her to dish out more damage, i suppose in some cases you could take Mobility Boots for more roaming potential but I peronally rarely, if ever, buy mobis.

The standard skill sequence I use for Katarina is as follows:
Death Lotus (R) > Bouncing Blades (Q) > Sinister steel (W) > Shunpo (E)
Obviously It's best for pretty much every champion to put points into their ultimate whenever possible, but I choose to level up the abilities that will give me the most damage output and this is achieved by maxing Q>W>E. There are some lane opponents that will use a shield to block the harass from your Q such as Kassadin or Riven and in these cases it can be better to max W first in lane.

On Katarina, being that she's a carry mid laner who specializes in killing people, I'm sure 90%+ of Katarina players, including myself, will tell you that Flash and Ignite are the way to go on Katarina as far as summoner spells.

Some other viable but "inferior" summoner spells:

Ghost:
Ghost has its advantages over flash in that you cover more distance but ghost really isn't the same kind of escape as flash and won't help you make the same kind of plays that flash will.

Exhaust:
Exhuast in a a way can increase your kill potential on a single target because it forces them to sit in your ultimate for longer but even with that it will not be as useful as ignite in all stages of the game.

Teleport:
Teleport is a great summoner spell to take if you find yourself playing Katarina in the top lane for some reason or if you have little to no confidence that you'll win lane.

Heal:
Heal is a decent summoner spell to take on mid laners if you want more sustain perhaps or are looking to bait the enemy laner.
All other summoner spells simply shouldn't be taken.

PROS:

Very High Burst

Instant escape/gap close

High AOE Damage

Great Team fighter

Can Make plays

CONS:

Very Squishy

Less Than Stellar Laning phase

Weak When Behind

Extremely vulnerable to crowd control

Semi-Reliant on Ultimate in Teamfights

TEAM FIGHTING WITH A HARD INITIATE:
Your job as Katarina in team fights is simple but not always simple to execute. Your job is to kill as many people as possible, prioritizing the enemy carries. In team fights you're going to want to abuse your passive as much as possible, if your team initiates the fight with hard cc such as Crescendo or Curse of the Sad Mummyon a large portion of the enemy team then Bouncing Blades> Shunpo> sinister steel> Death Lotus Q>E>W>R onto one of the carries and once a reset is granted Q>E>W the next lowest enemy and continue to use That combo to kill as many people as possible.

TEAM FIGHTING WITHOUT HARD INITIATE:
If your team doesn't have a very hard initiate then your going to want to wait for one of the enemy player to have low enough help that you can kill them with one Bouncing Blades> Shunpo> sinister steel (Q>E>W) combo then use the reset to dive the back line and use your ultimate until you can wipe the entire team with your resets. Even though your ultimate is the most damage that Katarina has in her kit, you're extremely vulnerable while using your ultimate because you're standing still and your Ultimate can be canceled by any hard cc. Katarina however is very hard to target and therefore very hard to cc and kill while she is shunpoing to fast for the enemy team to even see.

TEAM FIGHTING WHEN ENEMY INITIATES:
When the enemy teams initiates you should try your hardest to use Zhonya's Hourglass active Flash or a ward jump to escape and then if you think you can, you should go back in and follow the same recipe that you would use if your team had engaged, this in ways can be a very easy cleanup for Katarina because a lot of the enemy teams CC was likely used in the initial engage and this doesn't make you as vulnerable while using your ultimate.

Laning phase as Katarina is one of the most important parts of your game and is also the most difficult to play and takes a good bit of games to get a feel for. Your only source of harass in lane is your Bouncing Blades and this is one of the biggest down falls to her laning phase, use it whenever it's up and and remember that they do the most damage to the first target they hit. Until level 3 you shouldn't worry about trading at all as you'll get out traded by most other laners, a good example of one of these champions would be Fizz. You can really trade well with your lane opponent if you play correctly. If you're farming and the enemy laner comes in range for a Bouncing Blades then go ahead and use your Bouncing Blades > Shunpo > sinister steel combo and walk away the small speed bonus from your sinister steel and the small damage reduction from Shunpo should help you get back behind creeps without taking to much in return. The same Q>E>W combo is your bread and butter as Katarina so learn to do it quickly and get very comfortable with it. if you add your ultimate immediatley after this combo as well as an ignite on a semi-low lane opponent it will usually net you a kill. Also, if you think you have the damage it can be very easy to dive as Katarina due to the fact that you'll get the reset from a successful dive and be able to shunpo out of turret range with help from a ward or minion. You should always try to save shunpo to escape jungle ganks however, a shunpo to a ward or trinket ward or a allied minion can save you from a gank. Shunpo doesn't need to be used only to escape the jungler, you can always use shunpo to dodge the enemy laners skill shots. A lot of Katarinas matchups rely on her to dodge skill shots, using shunpo effectively and starting boots can help a lot with this.

The Number one unique skill i want to discuss with Katarina is ward jumping with Shunpo I personally don't buy sight stone on Katarina or anything and I'm not saying you should but i always like to have at least one Stealth Ward on me in addition to my trinket Stealth Ward to shunpo to. Play Video
Im including this video because in my eyes this guy truly is the best Kat NA but also because he has some stupidly quick fingers and can ward jump extremely well very quickly.

Thank you for reading my guide. I hope that you have as much fun playing Katarina as I have feel free to message me with any questions I'm Kickazz8 on NA.