Ok, I'm not going to say I'm an expert, I love ap carries and play almost always mid, but I'm a too skilled player. I'll just say what I think is best for this new char I instabuyed :P
First things you should know about ahri, in my opinion:
1. Her movespeed is 305. That's too low, specially for a char that needs to chase her enemies in order to finish them off (she has low cooldowns but not a very high damage). That's why I get movespeed instead of mana regen on masteries; also one of the reasons behind the lich's bane. You need to be fast and you are not (without your ult), so fix it.
2. Though most guides reccomend to max q first, I think maxing w first is way better. I will explain why in a while.
3. The built items are not the only choice, and a lot of different options are explained in the "different items" chapter, so don't just ignore this guide because there are no hp items in the main build or you don't like lich bane on casters.
4. You need to know when to attack. You cannot deal signifiant damege from afar, so you need to aproach you enemies, but you cannot rush on a group of enemies without harm. Use your ult for escaping or chasing, not for initiating (unless you want to enter, spam aoe and use your zhonya to get out, but it's pretty risky..)
5. Finally, as an item of note, with this build you'll have more mana than what is stated (the farming of tear of the godess / archangel's staff) and much more ap (as the passives of rabadon's deathcap (30% extra ap) and archangel's staff (3% of your mana converted to ap) are not counted by the mobafire calculator).

Pros: Incredible escaping mecanism with her ultimate.
Great aoe.
True damage on one of her skillshots.
Low cooldowns and mana costs.
New champ, so little people know well her skillset and how to counter it.

Cons: Not too high burst without throwing full combos.
Squishy (as most casters).
Too slow movespeed.
Mana hungry early on if playing too agressively.

Ok, these runes are, for me, the base of almost all ap champs. Some ap early on to harrass better, mana regen and magic pen. The only changes I think you would be wanting to make are movespeed quint to counter her poor movespeed, but that means 15 less ap, so make your choice. If possible I will add other rune options sometime.

The only difference from the normal caster masteries is getting movespeed instead of mana. I already explained why, so nothing to say here. aso you get 10% less time dead, wich is great late game.

This section is difficult. I have not tested Ahri enough to ensure this is the best item build, but I have played some games and it worked fine for me.

Why to begin with archangel's staff? Well, you need the Tear of the Goddess as early as possible to farm it and for that cute mp/5. You may start with health pots, but if you use well your passive you can pass without it. And once you have the tear of the goddess, archangel is the next item. You may want to get rabadons before finishing it, but I prefer it this way.

and Why take boots of speed so soon? One composed word: movespeed. I cannot state how much you need it in order to be a good Ahri. Also, once you get sorcerer's shoes 20 magic pen and even more movespeed is what you need.

Rabadon's Deathcap explains itself.

Lich Bane is a item I loved on her, even tough I do not use it in almost any other char. You have low cooldowns, many ways to get near ou enemy and so yo can proc it easily. That extra movespeed is also a good reason, as well as the ap and mana. It is also very easy to proc with your skillset; you have one CC that draws the enemy near you (giving you a nice moment to autoattack right after shooting your Orb of Deception) and a triple flash over 10 seconds, which gives you time to proc four times lich bane's passive, one on each flash and one more whith a cast of one ability (like your Fox-Fire). Also, as you can pretty much spam your W every 4 seconds with blue, you can get to proc the passive like hell for a fairly low mana cost. If you do not like this item, you may take Rylai's Crystal Scepter; this option will be explained in the next chapter.
Also remeber when building this item to start with Blasting Wand, as you won't get almost any profit from Sheen alone since your base AD is poor.

The void staff is pretty cheap, gives ap and more importantly magic pen. No more reasons.

As for Zhonya, armor is never a bad thing, and ap is even better. If the enemy is mainly physical damage dealer, take it. But the main reason for choosing this over Abyssal Mask is it's active. You ult in the middle of a team or an ambush? Zhonya. You are near an teammate who'll save you but won't have time to aproach him? Zhonya. And so on.

The deathfire grasp is also a good choice, specially against foes with huge nukes that you want to kill as soon as possible (veigar, for example), sice his active deals an amazing amount of damage. Also the CDR is very usefull, since you want your ult loaded everytime to be able to escape or finish people. If you go for this item, remember to take kage's lucky pick as soon as possible in order to take advantage of his gold over time (you probably may be wanting to take it as soon as you buy Boots).

This item is a good substitute of Archangel's Staff for two main reasons. First, it is the kind of item, as archangel's, that you want with you as soon as possible in order to get it farmed. Secondly, Catalyst the Protector is also a perfect substitute of the tear of the goddess, as it gives you survability while leveling up and lets you stay in lane longer. You get less mana and ap, and no mp/5, but the health is welcomed in a squishy that need to jump over her foes in order to fully damage them.
In case you prefer this item over archangel, you probably want to start with a Sapphire Crystal and one Mana Potion (and probably also a Health Potion) and then get catalyst, boots of speed, rod of ages, sorcerer shoes and then follow the build.

This is one of the recomended items for ahri, and good for the survability it gives and of course its passive. Also, it counters pretty well her poor movespeed: if you can't go faster, make them go slower. There is although one reason why I don't like it on Ahri: the only way to slow your enemy a 35% is with your e. But your e has his own CC. So yo can only slow a 15% in aoe, which is truly nice on teamfights, and can help you greatly when chasing an enemy, specially with a no-cc companion, but you are still not getting full profit of the passive. Just keep in mind your damage output will be much lower without lich's bane, so I would only reccomend this option if you are not dominatig or if you're having a hard time not dying.

This is a great option to take instead of Zhonya's Hourglass. Instead of armor it gives you mr, so if the enemy is heavy magic damage dealer, take this. Also, the -20 mr aura is a good boost to your damage, specially against foes who have litlle mr (if they don't buy mr at all, with this and Sorcerer's Shoes you'll be dealing true damage to almost all champs). And being someone who can easily approach enemies and then fall back, you can ult someone, throw a full e-q-w combo and then ult back if the enemy didn't die, and having all attacks take profit from the aura.

This one is a truly cheap item that synergizes well with your passive, giving you some extra ap and that cute spellvamp in aura. It greatly improves your survability specially if you cast orb of deception or fox-fire in a minion wave. Also is great when facing enemies with huge life but low damage output, as you will be able to keep fighting longer with almost full life. If your team is heavy ap and can benefit from the aura, feel free to take it, unless two or more of your teammates already have it.
Also, if you are having a hard time staying in lane, or facing heavy harrasment enemies (like Caitlyn) you can take it before Rabadon's Deathcap and sell it when making your last item.

I do not really like this item on Ahri, as the only real benefit (besides the ap, wich isn't too high) is the cdr, that in Ahri is only really needed for his ult. So unless you use too much your ult for attacking and so need it as soon as possible (in Ahri's case I recommend saving it only for finishing escaping foes, teamfights and specially as a life insurance), I won't take this one. Anyway, I'm big fan of this item with other casters, so I thought it deserved a place in the guide, as depending on the battle fought and the playstyle of the summoner, it may be a good option.

This item is great if you want to dominate through the entire game. Of course, you cannot take this item if you are not completely sure to be able to get a good kills/deaths rate. If you go for this option I recommend you to gank as much as possible for assists and kills in order to get stacks, and save your ult for escaping in order to avoid losing them.

In the case you are facing heavy CC enemies, you need that extra mr or simply ended your void sceptre and think you can pass on the sorcerer's shoes, go for this. Tenacity is the word. Anyway, I wouldn't take these ones as first boots unless you are having a truly hard time surviving enemy stuns, fears...

x2 This is the start I recconmend if you are aiming for Archangel's Staff as first item. If you go for this farm as much as possible and try avoiding recall at least until level 6 (or until you can get Tear of the Goddess and Boots). With the two potions, your range and your passive it should be easy to save those 955 gold you need if you do not go too agressively early and don't get ganked. It gives you enough mana to be able to cast a lot of spells both for farming and harrasing.

This is a good start and a standard for most casters. If you take this item aim for early kills in order to get enough money for your first recall. The extra ap and hp should help to this goal. It is also a good starting item if you want a [rod of ages] as first item, as it gives you no mp/5 unlike the archangel's and doran's ring.

x3 First of all, in this start you can perfectly get less health potions, or change some of them for mana potions if you prefer. It is a really good start if you want to gank early or if your opponent has skills that can be avoided, like ezreal, brand or morgana. It can help also greatly if your enemy is melee so you can kite easily.

x2 You can go for this if it is a component of your first item (like in the case of rod of ages). If you go for this I would recomemnd flat mp/5 seals instead of the other ones, in order to get extra regen when starting.

I don't like this start on ahri, but if you want to take will of the ancients early on or a mejai's soulstealer you can take it. But even in those case, I would recommend rathe a doran's ring. Anyway, it is a choice, so I list it here.

First of all, sorry for the icon, but as Ahri's passive name is the same of Nasus's, I don't know how to put her passive here. Ahri's passive gives her 35% spellvamp oh her next spell when charged. In order to charge it, yo need to land ability hits on enemies. For each hit, with a max of 3 per ability, yo get a charge, and on 9 charges you have it loaded.
It's a different version of morgana's passive Soul Siphon. You get a higher spellvamp but with a condition. At most, you can use it every 4 abilities, so it is kind of weaker, but enyway, if you take profit from it, you can get a really high survability in lane. Basically, you will be wanting to always use this passive with fox fire on minions, as it can hit 3 targets for a quite good amount of damage.

This skillshot deals some damage when going and some more when coming back. It deals the same amount of damage on the two hits, only that it is true damage the second time. True damage. As ignite. No magic resist to lower it. Early you won't notice much difference, but if you come across a 150mr shen you'll see how great this is. it has a pretty long and wide range, so you can use it for farming quite well (in some cases, better than fox fire) and stacking charges of your passive. the better moment to use it is after a charm, as there will be no way to dodge it.

In my opinion, the main Ahri skill. At level 5 it has 2 seconds less cooldown than her q and it's mana cost does not increase as you level it up. Also, although it deals magic damage only, it has a better ap scaling i used on a single target. This can be spammed while running from someone to hurt him, to farm minions if there is no champ too near and is the better skill for regaining life with your passive. Also the better for activating lich's bane proc, for it's low colldown and mana cost. If you want to hurt someone, spam it as hell.

A brand new type of CC. It's like a guided fear. Guided towards you. It deals a little damage and charms the enemy for 1 to 2 seconds. The range uses of this CC are wide. You can use it if the enemy comes near your turret to make the turret shoot him, you can use it to land an easy skillshot with your q and no chance of dodging, you can even use it as a little stun on difficult 1 vs 1. It's your better initiating skill, and so you must know when and how to use it. Be carefull though, it has long but thin range, and it may be dificult to hit targets from too far.

A triple aoe flash. At a ridiculous mana cost and at lvl 3 60 seconds cooldown. You can use it to storm trough teamfights dealing massive aoe with your whole set of abilities, to jump walls, to chase or escape or even to initiate if you are confident enough. It flashes automatically on where your pointer is, dealing damage to a max of 3 objectives in iits way. Remember to always use it's three charges before it goes on cooldown. If you just need one, you can flash back to where you came for and then reflash to the place where you initially wanted to go, so you deal three times damage. Also, do not waste it, it is perfect to run from almos any ambush you may fall into.

First question: why to start with Charm? simple: to punish early towerdive. Yes, it make easier to land both hits with your orb of deception and so on, but when you see an enemy come to you at half life early trying to finish you under your turret, you can get an easy kill with a simple skillshot.
Second: why maxing Fox-Fire first and not Orb of Deception? Well, lots of reasons. First, it has higer base damage and higer scaling if you land all three shots (wich is surprisingly easy). Also, obr's mana cost increases as you level it up, but w doesn't. It is great for both farming stacks on your passive and procing it if you use it on minions. And finally, it is not a skillshot, so it cannot be dodged. I think those are enough reasons to justify maxing w first.
There is not much more to say about the order, I'll update this section as soon as I learn more of Ahri's abilities.

Great options: Clarity, Teleport, Flash.
Clarify is great early on to stay in lane or attempt to finish off your enemy if low on mana. Teleport is one of the most usefull skills in game, for escaping, attacking, ganking, defending and almost everything.
Flash may be good for positioning in a teamfight before your ult, and also am great escaping tool.
Other options: Ignite and Exhaust for attacking, Heal for saving your life or helping towerdive, Ghost for chasing or escaping and Surge for the bonus ap.
I do not think any other summoner spell can be usefull on Ahri (and even think some of the "other options" are a wasted summoner spell, but anyway), but feel free to take what you wish.

In order to fully grasp the range of your ult, it may be a good idea to go on a custom game and use your ult just to jump around in the jungle, to see which walls can be jumped and which cannot.

As for comboing, the best way to do an ultimateless combo on an enemy is to land your Charm, follow immediatly with Orb of Deception so your foe gets hit twice, autoattack if you have lich's bane and Fox-Fire as last. If you think you can time it well enough and the enemy has no means of breaking your combo, like flash or stuns, in order to max the damage output, you may want to autoattack right after charm, then throw q and wait untill lich's bane passive can be proced again to use your w.

If you want to introduce the ultimate in your combo, just engage with it (being sure no ambush is set), autoattack, charm+orb, autoattack, fox-fire+ult, autoattack, ult out. Feel free to change the order of the skills in the combos, I just stated what seemed to work best for me, but everyone has his own gamestyle.

If you are chasing someone with your ultimate, try not to use all three dashes together. It is better to use your other spells (specially w, as you will almos always land the three hits automatically) and if you have lich's bane autoattacking while chasing them. Just remember to use your three dashes in the ten seconds timeframe so you don't waste any.

If you see someone trying to recall from his turret (which shouldn't be done if enemies are near) you have two options. One is to lure him with Charm as it has the longer range of your skills and follow with two fast w and q to end him, and the other, if you like looking cool and the enemy is at pretty low health, is to ult thrice over the range of the turret and escape by the jungle, as all three hits will land on the enemy and if you are fast enough you'll only take 2 (maybe even 1) turret hit. Of course, you can combine both strategies to almost instakill any tower hugger under 50% health.

When deciding between Abyssal Mask and Zhonya's Hourglass, remember to look not if the enemy autoattacks or not, but if it deals magical damage or physical. for example, teemo and warwick do mainly magical damage due to on hit effects and abilities, so against them it may be a better option the scepter and not the hourglass.

Little to say here. Just sorry for any faults I may have commited on my guide (I'm not english so probably I mispelled some words) and if you think you know something to improve the guide (there are still lots of work to do) just tell me. Also please, do not downvote without a comment on why you do not like the guide. I like constructive criticism, but not trolls (in fact I love trolls, but hate being trolled).