Katarina is a fairly simple AP champion who is versatile in both the lane and team fights. She is not a champion that can easily 1v5 for kills, however, she can take half the health of every champion in said 1v5. Once CC has been thrown, Katarina can jump in and grab several kills or assists.

Pros
-Easily learned
-Effective AoE nuke
-Built-in flash
-Powerful 1v1 champ
-Fun

Cons
-Squishy
-Usualy focused
-Made useless by CC

Flash

This spell allows Katarina more versatility, to escape ganks when no minions are near, or to fall back from a bad Shunpo

Teleport
Especially useful for the mid lane, Teleport lets you easily defend lanes, as well as return to your lane quickly after going to the shop.

I feel that Katarina has no real need to max any specific tree, but can utilize small parts of every tree. Offense gives her extra Mpen, Defense gives her dodge chance, and the Utility tree gives her improved experience, as well as help for her summoners.

Starting with an Amplifying Tome and HP pot is for more harassing power early game. Mejais should be grabbed the next time you go B, since you should usually have 10+ stacks. The boots are debatable, but I prefer the mobility from Boots of Swiftness. Rabadon's Deathcap will give you a very large amount of AP, and the 30% extra AP stacks well with Mejais. Rylais Crystal Scepter will give you a marginal amount of health and a decent amount of AP, but more importantly, will give all of your abilities a slow. This is perhaps the most useful item on your build, as it makes enemy champions eat more daggers from your ultimate. Lich Bane makes for some hilarious nuke damage with Shunpo, letting you finish off squishys with a Bouncing Blade followed by another proc. Banshees veil, if the game is not already over, can be replaced by any other defensive item. However, Banshees lets you eat a CC without cancelling your ultimate.

Voracity

Katarina's passive grants her extra gold when she gets a kill or assist, and reduces the cooldowns on all her abilities by 15 seconds when she recieves a kill or assist. When you are getting 2 or 3 kills/assists per team fight, your Death Lotus should be almost off cooldown after a team fight.
Bouncing Blade

Her main harassing ability in the lane, Bouncing Blade bounces off enemies, dealing 10% less damage with every bounce. Great for farming and harassing, bouncing blade shouls be maxed first.
Killer Instincts

The passive on this ability makes Katarina's other abilities and attacks deal extra damage. The active improves her next Shunpo or Bouncing Blade. If the next move used is bouncing blade, every bounce does the initial damage, and applies a debuff that reduces healing done. If Shunpo is used next, you take less damage for a short amount of time. Get a point in this early, but don't bother to get more until necessary.
Shunpo
This move works as an instant flash to any enemy or allied unit(not turrets). If it is an enemy, it takes a good amount of damage. It is great for escaping ganks, and jumping into a team fight to activate Death Lotus. I do not suggest using this often early game, the damage you will take will almost always exceed the damage you deal. If you are in the mid lane you might be able to do this well against squishier targets.
Death Lotus
This is Katarina's "I win button", it allows her to destroy enemy teams the way she does. Katarina throws many daggers over three seconds, htting up to 3 champions and dealing a decent amount of damage with each dagger. In a team fight this should ensure that your team wins without a problem, and in a 1v1 or 1v2 this allows you atleast force them to run. Level this whenever you can.

Katarina can take either mid or a side lane effectively. But, she is not a great character to go into a level 1 fight, but can easily harass from a distance. Try to last hit with bouncing blade, ensuring you are out of enemy harass lane. Also, the blade will often bounce to enemy champions. Using Killer Instincts before Bouncing Blade lets you deal maximum damage to enemy champions, as well as get more last hits. Once you hit level 6, they should have a decent amount of health taken, while you still have plenty. If Death Lotus can take them out, waste no time with Shunpo followed by your ultimate. Laning with a partner that has good CC is prefered, because it allows you to force all of your daggers at the enemy champion. After you kill those you are laned against, go back, or gank other lanes.

At about level 8, you should have a few stacks of Mejais, boots, and probably a needlessly largo rod. After this, you should never fully commit to a lane. Shunpo lets Katarina jump out of bushes and deal devastating damage. Followed by a Death Lotus, kills will come falling at your feet. You can choose to push towers, but I usually go to other lanes when the lane has been cleared, and let AD champs push the towers. Also, team fights should be starting. At this point, you should sit back for a second or two(let CC be thrown), Shunpo to a squishy on the enemy team, and hit R. Chase runners with Shunpo followed by a Bouncing Blade, as they should both be off cooldown from kills/assists. You should not get jungle buffs, and not try to solo dragon. Katarina is not built to kill and high health jungle monster with ease, and the buffs are definitely better spent on other champions.

At this point, you should have a large number of Mejais stacks. Also, you are most likely starting Lich Bane. Team fights are handing you kills if you have a decent team, and they should be pushed to their base. This stage isn't much different than mid game, just don't lose stacks and you will stay invincible.

Katarina should throw a Bouncing Blade at low health minions, preferably with an active Killer Instincts, to receive last hits.
Killer Instincts along with Bouncing Blade will cut off enemy health pot effectiveness, as well as other forms of heals.
Shunpo into a fight to get a quick kill, then Shunpo out to escape death.
Bouncing Blade may not deal large amounts of damage early game, but gradually it lets you take the advantage.
Play Katarina passively early game. The whole point of Bouncing Blade is to harass without being hit yourself.
If you are just out of Shunpo range, feel free to flash and then Shunpo for a quick chase.

Katarina is fairly squishy, but makes up for it with a great escape mechanism and tremendous amounts of damage. I am open to suggestions, please comment :)

Katarina is one of the most effective mid lanes in the game. Try not to lane against a vladamir when possible, he can pool out of your ult. Other mid lane champions, such as kennen, heimerdinger, anivia, ezreal, karthus, and others cannot win against a well played katarina. Also, if another katarina is mid against you, you will most likely have the advantage. A surprising amount of kats max shunpo first, which gives more damage, but is limited by range. Simply stay out of her shunpo range, and harrass her down with bouncing blade.