So this is suppose to be my introduction right ???
Katarina is an extreme and powerfull midler that can carry you and your team if you have some fast fingers and a good experience playing in the league. She has high mobility, tons of burst and aoe damage.
To start playing katarina you must also learn (in case u haven't yet) to play with smart cast keybinds, it makes all of your spells much faster to cast (if possible learn playing with smart + selfcast), giving you an enourmous speed advantage that is essential to this champion especially because of her passive ability (all spells, items and summoner spells must be smart cast). You will find it hard at the beggining, but once you've got into it, there is nothing like it !
Katarina, however, is very weak in laning phase, especially agaisnt poke and ranged champions, so you really must practice a lot with her in order to become patient (sometimes u just have to loose some minions, because if you get greedy and wish to last hit that minion in front of your opponent, he is gonna punish you hard, so DON'T GET GREEDY !!!), most of the times you have to be patient in not to waste ur cooldowns and wait to spam them on last hitting minions.
There are some tricks you can do to maximize your farming potential, such as casting w right after a melee attack in front of a minion about to die, melee attacking the non-focused minions so that after that you can blow three or four in one w rotation, and of course apply q when a minion is half hp in order to make the last hit easier. I don't reccomend you to waste schunpo on farm, unless you have the brushes warded (and I mean both), you know how it works, the jungler is always waiting.
There are some extra cares that I will talk about in laning phase, refering to playing agaisnt specific champions.
I hate mid game with katarina ! After you have grabbed some kills going botlane after getting ur ultimate ;) and after the enemy's mid turret is down or perhaps yours instead, everybody goes mid to blow up your damn farm and force you to go farm somewhere else. It is important to know how to play katarina, but it is also important, if not even more important, to know how to play with a kata that is your ally. It is a damn crime to steal kills to kata early game or farm, SHE NEEDS THEM !!! SHE DEALS TONS OF DAMAGE PEOPLE WAKE UP !!! SHE CAN NOOK DOWN ENTIRE TEAMS MOSTLY ALL BY HERSELF !!! Junglers, AD carries, Top laners and most funny supports should know that and always remember, when you steal her farm, take her kills and don't gank her lane, you are playing for the enemy team.
Another problem is cc, which I don't really consider it a problem at all, but that is because of my playing. I will discuss it later on more specificly, but mainly what you need to know is that you can't just dive in e+q+w+r like boom. There is a timing for you to enter the fight, just like the budda karate stuff where the guy meditates for two years and then suddenly wakes up and says: it is time !

Katarina needs the flash, it is most necessary to either engage, disengage, simply escape or kite. Yes ! Katarina does kites !
Ignite is very good when u combo it with hextech and ulti, it counters hp gains which makes it essential agaisnt some opponents.
Teleport should be the second option besides ignite, I only reccomend it playing all premade, because you never know when you are going to use it for nothing.
Barrier might be wierd, but agaisnt champions like fizz, malzahar, diana, cassiopeia, veigar, akali and ahri can come in handy since they are highly superior in lane, and you can forge a mistake in order to kill them with your turret using barrier. It's lame but it works !
Ghost if you are greedy xD, Im kiding I never used ghost.

As I start out with low ap, I take the tome and only one pot in the beggining, it gives me low sustain, but if you play right you can stay in lane until you get around 850 gold (enough to get spellvamp, pots and at least one ward). The tome is necessary in order to make your early farming easier. Please don't take dorans !!! Just don't !!!
I rush the hextech of course, and I put the boots after it because I don't really buy them unless I am ganking the botlane (I rarely gank top, as it is more dangerous when counterganking and the top laner is usualy harder to bring down). Sometimes I just buy them after having the Lich Bane.
I forgot saying that the hextech is essential on Katarina and that no other spellvamp item should be taken as a replace (it boosts your ultimate, gives you higher lane sustain, more burst and the slow that works super with the ultimate).
The Lich Bane, this might be hard to understand, but it works very very good, especially agaisnt your biggest counters and when kiting or chasing tanks or other targets. It is great for turrets (as well as your ad runes ;) ) it procs with ur q, gives you extra speed to escape or chase, increases your burst on single target damage and most importantly, breaks the mith that katarina is ulti dependant (it also procs with your cooldown boots). Never felt out of damage after using ulti on a target and you are feeling powerless agaisnt the enemy adc that approaches because u have no ignite or abilities to spam, the lich bane might not solve the problem at all times, but it has saved me through many.
The cooldown boots omg this guy is a noob, did he read kata's passive ??? is he ****** ??? probably yes. The fact is that it has proven to be way better. Have you ever died knowing that you could have killed the target but the ability was 0.5 secs from coming back, and you think, damn I should have bought the cld boots. Kata needs cooldowns, Kata needs blue, as adcs need attack speed to make use of their ad, ap carries need cooldown reduction to cast faster. The magic pen boots are way more expensive and only give 15 magic pen. What does 15 magic pen does on a tank ??? Nothing !!! and how bout 15% cooldown ??? magic penetration is always better in percentage, because it works both ways. Trust me, Kata has enough damage to nook an adc or a midler, what she hasn't is damage enough to kill a tank or a fed bruiser.
If you are in a team fight and trundle tank rushes you, making it impossible for you to engage the rest of the enemy team, will you walk through him and risk taking damage, or will you simply kite him and deal tons of damage on him ??? This is where lich bane and cld enters. You cannot waste ulti on him cause it is simply not worth wasting on, so you have to damage him with what you have, your passive will be useless, so you have to make use of your cooldowns. Be Smart !!!
Zhonias is as must have, u can use it to reset clds, avoid abilities, get some extra time, whatever.
Void staff is alternative as well as lich bane althout all I said before. Zhonyas, abyssal scepter and deathfire grady and other great items to use on her. I only use the staff when I either want to nook faster, or simply because the enemy team has way to much magic resist. I take Rilay when I am dying too fast or when I feel the need to kite a bit more, and deathfire grasp, well, I rarely use it, but it is a great alternative to lich bane and works great on any target. Remember that raising the damage dealth also increases the ammount of health that you gain from spellvamp.
I obviously use the abyssal scepter when the enemy team has either a very strong midler, when the enemy team has tons of people dealing magic damage, or, if my team has way to much people dealing magic damage. If you buy it because the enemy team has a strong midler, than you must have in consideration that magic resist by itself is not worth buying if you don't have a good hp, so in that case, I reccomend you to take hpprlvl seals and to buy rilay. More defensive is kinda gay xD.
And finaly and probably the most important, the wards. Katarina don't usually needs to have the brushes warded, you don't really need to ward at all, you need to understand how to use wards as an utility item to shunpo on. If you don't have enough gold to buy two wards for both brushes, or if you simply are feeling that you are wasting much money, or if the enemy is a fizz for example, you can ward the lane at the mid point when you are pushing or simply save the ward to use it when u need to. Wards can be used to approach targets that you are chasing, and although u might have spent shunpo on it, you can after use the hextech to slow the target and catch him up !!! And that is when the lich bane and the cooldown reduction will be most benefitial, because the burst damage will be higher as well as the speed, if u didin't dealt enough damage to kill the target, don't worry !!! your w will come back very fast again, and with him, the lich bane. Just think about it.

I always start with q no matter what, and I never help the jungle pull in our side so that I can start pushing before the enemy midler, that way you can reach lvl 2 faster and take some advantage in lane (that kata so desperatly needs!!!)
The second spell, althout I selected w, is relative. If you are feeling too harassed, or you are simply afraid of a lvl 2 gank, I suggest taking in schunpo.
After that, I think that it depends on the playstyle, but as I play kata very defensive, I reccomend to get at least the rank 2 on the w spell in order to last hit better. schunpo is lvled sometimes in priority to q which I don't reccomend at all, because as an ap midler, I think that your job in a tf is to deal the maximum possible damage, and it won't be with melee attacks.

The Masteries, some people find it stupid, I find it to be my masterpiece, and my advantage playing Katarina.
But of course, it is obviously very basic, the main difference is in the attack damage (Katarina's ultimate ability procs 3 times more from your bonus ad) and it also scales with your w, good to farm anh ?
The archmage is the only thing I regret not having, but it simply isn't worth going for, in early game I have apprlvl runes, and my first items are not really strong ap items (I will actually have more ad than ap for a while), so It is only worth having just for late game.
Some people take preserverance, but I don't use it because I rush sustain items and buy tons of pots early. Don't get a dorans on her please, don't do that ! The hpprlvl is better and if you play right you won't need that masterie at all, not talking about that it is practicly useless late game.
If you know you are playing agaist an ad midler, I suggest making an extra page almost equal changing the armor value for the magic resist one. Be smart !!!

The Runes I think it will be where I will be most criticized.
The only thing I have to say is try it, and of course, as kata is very weak early, there is no point in trying to make her look what she is not. You can rarely get blood on the enemy midler with early game runes, so I take late game ones. You only have to wait until lvl 8 (approximately), and after that it's all benefit.
Many people find attack damage per lvl wierd, I don't really care for any comments, just try out and then thank me. The optional seals are hpprlvl, and the optional quintecensses are spellvamp. I find adprlvl quints way too exagerated, u get more profit out of apprlvl.

It is very important to understand that kata is not a strong 1v1 champion compared to other midlers, such as fizz and diana, who's damage is mostly 1v1. You have to remember that even fed and the enemy not, he might still be able to win in an all in trade. They have more natural hp, more 1v1 damage, etc etc...
You should not demoralize when you die in a ganking situations, or even vs your opponent (remember that if you die in lane 1v1, it is only because you are stupid and greedy, or because you got tower dived, but it usually happens because you simply just make mistakes). You have to think that your advantage is being one of the ap carries with the biggest aoe damage. Your ultimate ability is probably the biggest burst damage aoe there is, and that is your advantage.
In my opinion, I think that aoe damage is the most important factor in ap carries, that is why I don't give credit on champions like fizz and diana, cause their damage is mainly focused, and agaisnt teams with either tons of aoe damage or hp, they won't just stand a chance.
Dying on a trade at early game envolving many players shoudln't be a concern. You must always remember that your runes are ment for later on, and that when the enemy team makes mistakes, kata will have no mercy. So always be positive about that, don't bother starting with a bad score, you will eventually get fed.