When I watched the Champion Spotlight on Orianna, I thought that she might be a champion I'd like to play. So the first thing I did when I logged on after she became avaliable, was purchasing her, and playing a custom game with her. And it seemed like I were right, because I found her ball mechanic very funny, and I'm used to playing Urgot, so I was quite fond of her ability to fight at a long range.

Lore:

There once was a Piltovian man named Corin Reveck who had a daughter named Orianna, whom he loved more than anything else in the world. Though Orianna had incredible talent for dancing, she was deeply fascinated by the champions of the League of Legends. This fascination compelled her to begin training to become such a champion. It is unfortunate that her sheltered, wide-eyed naivete led her to take unnecessary and dangerous chances which ultimately led to her tragic demise. Orianna's death shattered Corin, driving him into deep depression and an obsession with techmaturgy. He could not stand the void his daughter's death left in his life, so he decided to build a replacement – one that would complete Orianna's dream of joining the League. What was created is the clockwork killing machine that Corin named after his daughter. Knowing that she was destined to be a champion and seeing the way the times were changing, he created The Ball as her pet and protector. This nearly symbiotic creation uses a different type of techmaturgy, relying more heavily on electricity than clockwork.

Orianna and The Ball now fight as Champions in the League of Legends, using the Clockwork Girl's sometimes misguided morality as a compass. She tries in earnest to fit in with those around her. However, no matter how hard she tries, Orianna can never be human and there is always something unnerving and alien about her. Though she attempts social interaction with other champions in the League of Legends, there are few who can get past her exotic nature. To many, it's as if there's nothing inside, that Orianna is just a soulless clockwork shell – a dangerous and deadly one at that. However, all along she remains the perfect daughter in her father's eyes.

“Dance with me, my pet. Dance with me into oblivion.”

Passive – Clockwork Windup:
Orianna’s autoattacks deal additional magic damage every hit, subsequent attacks on the same target within a few seconds will add more damage per hit. This bonus stacks up to three times.

Command - Attack:
Orianna commands her ball to fly towards target location, dealing damage to targets hit but doing reduced damage for each additional target hit. Her ball remains behind at that location afterwards.

Command – Dissonance:
Orianna commands her ball to emit a magnetic pulse, dealing damage to units around it. Afterwards, the ball leaves a field behind for a few seconds that speeds up allies and slows enemies.

Command – Protect:
(Passive): The allied champion the ball is attached to gains bonus Armor and Magic Resistance.
(Active): Orianna commands her ball to fly to and attach onto an allied champion, dealing damage to enemies it passes through and shielding the allied champion when it arrives.

Command – Shockwave:
Orianna commands her ball to emit a shockwave after a short delay, flinging affected enemies in the vicinity into the air a set distance towards, and possibly over, her ball.

Pros:
Long threat range
Low manacost
Able to attack enemies without being near
Fast get-away options
Can scout with her ball
She's GLaDOS reincarnated, 'nuff said

Cons:
Not too much health
Mana issues early on
Sometimes you'll lose track of your ball, and you'll die horribly for it. Remember; When you've used Command: Protect on someone else, using Command: Dissonance won't speed you up!

To be honest with you guys, I don't use runes, I'm not even 30 yet, but this is the runes I'd use if I could.

Greater Mark of Insight
I picked these because... Well, we're playing a mage, we want magic penetration so we hit harder :)

Greater Seal of Replenishment
Flat mana regeneration? The reason for this is that I've got mana problems with her early on, but rarely in the late-game.

Greater Glyph of Force
Again, we're playing a mage, so we want ability power too :D

Greater Quintessence of Swiftness
These'll make it (slightly) easier to hit the enemies, and will of course make it easier to chase/escape. And with Quickness and Command: Dissonance we can get away very quickly.

Of course, this is just how I'd build her, and I'm sure that there's tons of other ways to do it, and probably some which is better than this :P

I buy Sapphire Crystal first, in addition to two pots to keep me in lane for as long as possible.
After that, I buy the Boots of Speed (Of course :P), and then get Tear of the Goddess, followed by Catalyst the Protector, and upgrade that to Rod of Ages as soon as possible, in order to gain it's benefits quickly. Then I upgrade my Tear of the Goddess to an Archangel's Staff, because, well, we cast a lot of spells, so we won't take too long to fill it up.
When I've got those items, I go for more AP, because I play Orianna as a mage, if I wanted to play support, I'd go for someone like Sona :P

I'd say that Rabadon's Deathcap and Void Staff are fairly stable, but you might want to get some items which gives you more survivability, should it be if needed.

I choose Clarity because, as I said before, I tennd to run OOM early game.

Blink is just a favourite of mine, and I can't count the numbers of times it've saved me.

Of course, these are just personal preferences, and I could easily see the reasons for taking other summoner spells such as Teleport or Ignite

Farming: Well... Orianna is very easy to farm with, in the mid-game, Command: Attack and Command: Dissonance'll be enough to finish off caster minions, and Command: Attack should be off cooldown in time for you to use it to finish off the melee minions.

Harassing: This is pretty easy too. You can tag an ally (Preferrably a melee with charge/pull) with Command: Protect and use him as a beacon for Command: Dissonance, or to get a better chance to hit enemies with Command: Attack.
Generally just use Command: Attack whenever you get the chance to hit the enemy. Personally I just use Command: Dissonance to slow down the enemies in early-game, but you may want to put more levels in it in order to make it good for damaging too.
If you've pushed to the enemy turret, shoot the ball a little in front of their turret, so that when the enemy champions move forward, you can easily hit them with Command: Attack.

Teamfights: Well... This isn't really about in which order you use abilities, just make sure that they hit the enemies. But I'd suggest saving your ulti for when you can get all of the enemies, or when you can interrupt an important ability, like Karthus' or Nunu's ulti.
Of course, remember to use your Command: Protect on allies who're taking damage, preferrably so that you can hit enemies when you do so. But it's better to use Command: Protect and Command: Dissonance to save an ally, than "just" to damage the enemies.

Something which I feel is an important part of playing Orianna is how you position your ball, and how you use this positioning.

First, remember that the ball gives vision around it, so you can use it to scout bushes.

I tend to throw the ball in the opponents bushes in the start of the game, and then trying to hit them once the enemies come in range. When harassing, you should also aim Command: Attack behind the enemy if possible, that way you might be able to use Command: Attack to hit the enemies without actually being in range of them.

I like to have the ball right in front of the enemy turret when I'm pushing, so that I'll be able to use Command: Dissonance to damage enemies which gets too close to the ball, or Command: Attack, should they come between the ball and the range of the ability :)

So this is how I build Orianna (At the writing moment, that is), and some of my thought on how you can be as annoying as possible :)

If there's any spelling errors, please bear over with me, english isn't my native langague :)

I hope to post some pictures soon, as I do realize that it probably looks a little boring at the moment :P

3/6: Minor changes to the guide, will probably do this some times over the next couple of days