It is important to know that Katarina is a champion that has a lot of potential, especially mid and late-game. Throughout this guide, I will be explaining the more minuscule concepts in as much detail as the larger ones. The first questions you must ask yourself are "What are her strengths and weaknesses?" and "How can I make the most optimal use of this champion?" The most important thing to realize is the role she plays within her team comp.

Katarina is a burst mage, though her auto attacks are melee. The main role that she plays is cleanup-duty. When a kill is left unfinished, or the enemy team is generally low on health, she is the perfect champion to finish them off, avenging her teammates, and sometimes saving the day.

Strengths:
-useful in team fights
-can easily take on more than one opponent by herself
-no mana usage
-good sync between her passive and her skills
-good escapability (even without escape spells and can Shunpo to sight wards)
-can be placed in more than just one lane
-ultimate syncs well with other aoe (area of effect) or disabling champions
-her ult has a short cooldown
-she is easy to get multiple kills with
-her ult can detect stealthed or non-visible champs when off cooldown

Disadvantages:
-her ultimate can be interrupted by other non-ultimate skills
-her ult can be avoided (especially by flash)
-not the best farming skills, but okay for last-hitting
-likely to be focused in a team fight
-vulnerable when abilities are on cooldown
-cannot push lanes quickly

These are a bunch of things you may want to avoid when playing as Katarina.
-not farming
-being too aggressive early-game
-being too cautious (but with this build, you will be prepared for many situations)
-bad positioning
-using her ultimate too late
-ganking too much to where you fall behind
-Using her ultimate to initiate
-bad ultimate placement
-premature ultimate

1) repositioning her at the center of your team
Why: Even though it may seem to be a good idea at first, you must remember that she can quickly get out of the situation, or even randomly ult at the center of your team. Even in a team fight where there is a numbers advantage, you may not want to get your team at low health at the beginning of the fight.

Singed - Can fling her out of her ultimate/because she gets most of her kills from chasing, she is especially affected by his poison.

Shaco - His boxes stop her ultimate if she is feared/his clone does much faster than he does, so she can tell exactly which clone is the real shaco, however, if the clone explodes, she could die immediately, or even put herself at greater risk after ulting

Valdimir - His pool allows him to completely dodge her ultimate

Volibear - Can fling her out of her ultimate

Singed - Can fling her out of her ultimate

Alistar - Can cancel her ult and reposition her so that the enemy team is safe or so that she is in danger

Warwick - Infininate Duress stops her ultimate/if he survives her ult he can still kill her if she is unable to burst him down before he life steals everything back.

Veigar - Event Horizon can prevent her from both escaping and chasing/his ult will hurt!

Nocturne - if he can get his fear off during her ultimate, she's screwed/his shield can save him from Shunpo, and Bouncing Blade.

Fiddlesticks- Not only will his fear prove fatal, but because he can silence her, she can be instantly shutdown early-game, and absolutely destroyed late-game.

Shyvana- Her ultimate can disrupt Kat's ult, and also reposition her in a disadvantageous place/ she can chase down Kat and finish her off/She may be too tanky late-game for Kat to burst her down right away

Nasus - Can get very tanky and hard to kill/ his slow can remove her ability to finish off kills in a team fight

Kassadin- His silences can prove disastrous, preventing her from taking any actions at critical moments.

Urgot- he has plenty of early-game harass and poke/late game: Katarina is a late-game champ, and urgot scale better early game (see the problem here? :P )

Lux- if she can bind you, you will die, unless your heal and/or flash are up/late game: If you can stay alive, and dodge her Light Binding (with Shunpo, especially) you are likely to get a kill off of her. Plus, she will eventually have to go back for mana, meanwhile, you will not, and your health will regen most harass by the time she gets back.

Ryze- his ultimate will give him free spell vamp, so until Kat gets her revolver, she might be at a disadvantage early-game./late game: Ryze's Rune Prison will not be very useful when Kat's ult hurts the most, and of course he mus be in range of her ultimate to cast the spell. If anything, his Rune Prison will be able to save lives, and put her in danger after ulting (when either she wants to escape or pursue).

Gangplank (Critplank)- Early-game harassment is hard to deal with, so don't plan on winning this matchup, unless you can stay alive for all of early and mid game/late game: he will melt under Katarina's "Spinning Fury" (unless he builds tons of magic resist, and/or is fed).

Ziggs- His early-game harass may be too much for Kat, if she doesn't dodge enough of his skills/ satchel charge can disrupt her ultimate

Cassiopeia- she has much more harass than Kat can handle, so it is best to turret hug/ her ultimate disrupts Kat's

Brand- he dominates the lane early game, when he harasses, in a way, he is very much like Cassiopeia, but with a more frequent stun.

Heal- Ah, the spell that beings out the word "noob" more often than not. But of course, Heal is an excellent spell to use on Katarina, due to the fact that she often uses her ult in the center of the enemy team. While your opponents are focusing you, a heal would quickly make them realize that they've been wasting their time, and that they should have focused the other damage dealers on your team. Also, this spell prevents you from dying to champs with binding spells such as Lux, Ryze, Swain, Morgana, etc. This spell will also help keep you alive until you get your gunblade. ^_^

Flash- More often than not, you will use flash for closing a gap, enough to finish off a kill with Shunpo. It is also a useful tool in dodging other fatal aoe spells and ultimates, then countering with your own once it has gone off.

It is important to remember, that Spell Vamp is her best friend, for it not only provides damage output, but a free heal. Also, the extra magic penetration should hurt your opponents even more. Replacing magic penetration runes with ability power runes, should also prove very effective.

The goal is this mastery setup is to maximize damage output, and the effectiveness of your summoner spells. Executioner is the most important investment because the intent is to kill your enemies before then can gain back enough health to actually fight you.

Bouncing Blade should have top priority out of all 3 of her non-ultimate skills. This is your main farming skill, allowing you to last-hit minions (and sometimes get multiple kills). This skill also scales with damage, allowing you to benefit even more from Hextech Gunblade.

Death Lotus the the skill that will make you a powerful presence in team fights.

Doran's Shield- Now, you're probably wondering... why would you ever get armor when you are likely to fight against magic damage in mid? Most champions who you will go against early-game will quickly get you at low health before even taking some harass themselves. By turret hugging, you might be able to regenerate a significant amount of your health before level 6-7, where you should be a more worthy opponent.

On your first trip back to base, you should have enough money to buy both boots of speed and a Hextech Revolver. The extra movement will allow you to chase your opponent more successfully, and the Hextech will provide you with just enough spell vamp to either get out of sticky situations, or stay alive in the early team fights. Before completing it into a Hextech Gunblade, it is important to pick up a Rylai's for both the extra health and damage.

I recommend Sorc Shoes over Boots of Lucidity for two reasons: 1) You want to be able to melt through as many champions on the enemy team as possible. 2) There is no point in having cooldown boots when all of your abilities (not including your ult) refresh upon kills or assists. If anything, you shouldn't be chasing, for long distances, but rather finishing off kills that your teammates could not and assassinating unsuspecting foes.

I always build a Hextech Gunblade over a Will of the Ancients. It is important to remember that Kat's Bouncing Blade and ultimate stack with physical damage and Magic damage, so it makes more sense to buy the Gunblade. Also, the active on the Gunblade will allow you to save teammates, save yourself, and secure kills.

When building a Rylai's, make sure you prioritize the Blasting Wand and Giant's Belt, depending on the enemy team. If you are up against a squishy team, you might want to go for the Blasting Want first. But if you're having a rough early-game, and are too squishy to stay alive in some of the early team fights, you may want to pick up a Giant's Belt first.

Situations Items :

A Zhonya's should be bought if you are facing enemies that easily counter you, or disrupt your ultimate. You should also buy this item if you are being harshly focused in team fights.

If the enemy team is tanky or with plenty of magic resist, you might want to invest in a void staff.

A Quick Silver Sash (QSS) is a great item against an enemy team with lots of CC, or someone like Nasus who's slow can immediately shut you down. Its a great escape tool, which gives you the chance at doing what you intended once you've entered a team fight.

An Abyssal Scepter is great defense against a mostly magic-damage team. Not only does it protect you, but it makes you hurt them even more.

A Death Cap is the perfect item when facing enemies who only build offensively. If you do not feel that you are in any specific type of danger, you might want to invest in one. At the same time, its a great item to buy when you "don't know what to buy last".

This video is a perfect example of how you should enter a team fight as Katarina. Play Video
This video can show you examples of how to juke and escape enemies in tough situations. Play Video
Here is a perfect example of both! Play Video