So ... I normally don't write guides because it is my belief that telling someone how to play something is 1) boring, because it's more fun and rewarding to figure things out for yourself, 2) wrong, because there are many different ways to play, all of which can be equally "viable," and 3) stupidly rigid, because people tend to follow guides to a T without considering the situation. However, some of my friends have been bugging me to write one, so I'll give my perspective on Ahri.

This is my first guide. Yes, the formatting on this guide is ugly. I'll clean it up as I learn BBCode.

I'm also assuming a basic level of competence in farming, map awareness, and knowledge of laning against various champions.

Pros

• Good damage output throughout the game
• Easy to last hit with
• Mana isn't a big issue
• Near impossible to gank
• Near impossible to run from
• Can reposition herself as needed
• Excellent poke
• Charm wins games

Cons

• Squishy
• Slow (dat 305 base ms)
• Requires GOOD POSITIONING
• Requires LANDING SKILLSHOTS

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Reds
Greater Mark of Magic Penetration - Standard AP marks

Yellows
Greater Seal of Scaling Mana Regeneration - Mana regen, flat or scaling, depending on when you feel you need it the most. Flat for early, scaling for late.
Greater Seal of Vitality - If you don't need the mana regen, health per level works nicely.

Blues
Greater Glyph of Scaling Ability Power - AP per level, standard.
Greater Glyph of Magic Resist - Flat MR might be useful against burst champions like LeBlanc, who you do not want to feed.

Quints
Greater Quintessence of Ability Power - Flat AP, standard.
Greater Quintessence of Health - Flat health if you want to be more tanky (helps early game a lot).
Greater Quintessence of Movement Speed - Movespeed if you can't stand Ahri's slowness (me; these things are good on most anyone anyways).

Standard 21/0/9 for APs. Mana masteries in utility is obvious, but you can take ms if you prefer it.

　
　
Ignite - Secures kills.
Flash - Derp.

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Exhaust - Viable if you team isn't packing one, but I don't recommend it otherwise.
Garrison - What is Dominion?
Ghost - Viable if you don't feel a fourth flash is necessary. However, I personally find flash allows for some amazing plays (counterflash, tower dives, etc.) without having to burn a charge on Spirit Rush (which also wastes damage).
Heal - This thing is good on anyone. There are better options. Meh.
Revive - No.
Smite - Jungle Ahri? k.
Surge - Usable? Yes. Are there better options? Hell yes.
Teleport - This thing is good on anyone. Meh.
Cleanse - Good against heavy CC teams, since Ahri explodes upon being stunned and silence makes her weep.
Clarity - FFS, you have tons of mana regen options in runes and masteries. If you're still running out of mana, spam less. There are better options.
Promote - Lolno.
Clairvoyance - Go ahead if you're support or have some obsession with facechecking bushes without it. Otherwise, no.

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Essence Theft - Built-in spell vamp. Handy passive, but it requires you to spam your skills and/or push lane.

Orb of Deception - Good for farming and early harass. Takes a nice chunk out of enemy health if you land it on the sweet spot (very end before it starts returning) or just get both ends of damage; lackluster otherwise. Plus it's scaling true damage. Who doesn't like true damage?

Fox-Fire - The range at which it auto-targets champion is actually not that long, so it requires you to be fairly close to your target if you're attacking with this. Spammable with low cooldown and constitutes a large amount of damage mid/late. Each fireball also counts as single-target (explained later).

Charm - Basically a taunt+fear combined into one. Neutralizes dangerous targets for a short amount of time and gets them to move into range of Fox-Fire and stay still long enough for Orb of Deception to hit twice.

Spirit Rush - The only mobile thing about Ahri, but 3 flashes makes lots of people QQ. Use this to burst, escape, reposition, chase, etc.

　
　
Max Orb of Deception for farming and poking. Provides the best heal with your passive early.

Max Fox-Fire second. It's possible to max this first if you're against melee, but it's rather unreliable when there is a minion pack. Orb of Deception also does more damage overall (AoE).

One point in Charm and max it last. 1 second is good enough to screw people over early, so there's no reason to max this before Orb of Deception or Fox-Fire. Take it at level 2 or 4, depending on the enemy jungler -- e.g. take charm at 2 against potentially early gankers like Lee Sin, Rammus, Shaco, Twitch, etc. or delay it to 4 against late gankers, which is pretty much everyone else. If you're not sure, taking it at 2 is always the safe route.

Spirit Rush is an R. Enough said.

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Boots + 3 Health Potion, standard opening. Movespeed helps escape jungle ganks and dodge skillshots, pots let you stay in lane for as long as possible. Mana costs on spells are low early.

Here's why you don't take Doran's Ring first: look at your ms. 305. Now look at your ms again. 305. Look at your Doran's Ring, and then look at your ms again. THREE OH FIVE. GOOD LUCK ESCAPING JUNGLERS WITHOUT BURNING FLASH EARLY OR DODGING SKILLSHOTS, AND GOD HELP YOU IF YOU'RE LANING AGAINST Cassiopeia.

Doran's Ring gives everything Ahri needs: health, AP, mana regen.

Rylai's Crystal Scepter gives good AP and health so she won't explode upon getting close to a target with Spirit Rush. The best part about this item is the slow, and this is why I prefer to rush it. Fox-Fire hits up to three targets, but each one counts as single target, giving you max slow on a very low cooldown. Thus, when you're trying to escape or chase someone down, pop Fox-Fire and watch the magic. If you're at max distance on foxfire, the spacing between each fireball will allow for about a 3 second slow, by which time your other skills will be up to slow some more, plus another foxfire.

Upgrade your boots whenever you can, preferably on your first back with doran's rings.

This is where item paths will depend on the situation. ALWAYS CONSIDER YOUR SITUATION IN BUYING YOUR NEXT ITEM. Otherwise, the item path I listed at the top is my preferred build.
　
　

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Will of the Ancients - Not necessary because of your passive. However, it's cheap AP and since Foxfire counts as single target, it provides massive heals. Late game, you can take a lot of damage and stay in fights by spamming Foxfire. Rush Hextech Revolver over Rylai's if you need extra sustain in lane.

Rabadon's Deathcap - The AP version of Infinity Edge. Get this at some point or do laughable damage.

Abyssal Mask - Need MR and AP, and the enemy team isn't packing tons of MR? This is your item. Switch this out with void staff if they do stack MR. Situational.

Zhonya's Hourglass - Need armour and AP? This is your item. The stasis is also nice if you find yourself being focused in fights. Pop it and watch the enemy team redirect their attention elsewhere. It also makes Karthus rage by dodging his Requiem. Situational.

Void Staff - See Abyssal. Situational.

Deathfire Grasp - AP, mana regen, and CDR is always useful. This combined with blue buff, blue pot, and masteries gives max CDR. Active is nice too. Switch this out for Morello's Evil Tome if you don't like actives or prefer the extra AP and mana regen. Situational.

Morello's Evil Tome - See DFG. Situational.

Lich Bane - If the game is dragging out that long, go for it. Procs often with foxfire at max CDR. Situational.

Banshee's Veil - Good stats all around. Kind of costly and the bubble is ineffective, but it can save your life against champions that need a full combo to hurt you (e.g. AP Sion, LeBlanc). Situational.

Quicksilver Sash - This thing is awesome. Enough said. Well, no. If they have anyone with suppression, you want this. IF YOU ARE LANING AGAINST A Malzahar, YOU GET THIS NO MATTER WHAT. Situational.

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Ionian Boots of Lucidity - Admittedly, I have not tried this. I can imagine it working because of the true damage on Q, but I normally get max CDR (or close to it) anyways, making this a waste.

Archangel's Staff - Ahri is spammy enough that you can max the mana bonus on this thing, but I feel there are better options. You also don't really need the mana and this does nothing else for you. Meh.

Guardian Angel - Good on anyone, but extremely situational. The nice side to it though is that people tend to avoid targeting someone with GA because they think it'll be a waste of time.

Mejai's Soulstealer - Oh looky, a snowball item, the very definition of situational. Incredibly unreliable in my opinion, but take it at your discretion.

Typical lane phase. If your jungler is going wolves --> blue, pop a Orb of Deception on both so you start with 6 stacks on Essence Theft -- you can actually do the same with wraiths --> red. Just make sure the jungler attacks the wolves/wraiths first so you don't drag them out.

Ahri's attack animation is fast and smooth, so last hitting is easy. Focus on farming, and throw out a Orb of Deception to catch cs you can't reach or hit on time (basically double-attack). DON'T SPAM YOUR ABILITIES TO FARM IF YOU DON'T HAVE WARDS. If your opponent freezes the lane near his turret, you put yourself in danger of a jungle gank to get cs. This is why I say you don't need much mana early.

A useful trick is to use rank 1 Fox-Fire for passive stacks, and then Orb of Deception on a full wave for health. The pushes your lane the least without taxing your mana while giving the best heal.

Ahri can be a bully in lane. Whenever your opponent goes in to last hit, throw out a Orb of Deception at him immediately. Move as needed to get both ends of damage on him. If he's juking off on the side to avoid being hit by Orb of Deception, move close and pop Fox-Fire so all 3 fireballs hit him. Use a full combo of Charm+ Orb of Deception+ Fox-Fire when you can (near full mana and/or opponent is low health). Your goal is to gain a gold advantage while harassing your opponent low enough to get bursted by your level 6 combo (R+E+Q+W+Ignite+R+R)(provided they aren't stupid enough to die sooner).

Gank side lanes when you can.

What's the difference? Keep farming, roaming, ganking, pushing other lanes down, securing objectives, etc.

Another tip: If you think a bush might be dangerous to facecheck, throw a Charm into it. If there is someone in there, you'll hear the smack. Alternatively, you can throw an Orb of Deception instead. If you gain passive stacks, you'll know someone is hiding in there. If you gain the full 3 stacks, then there is more than one enemy hiding there.

Pre-fight Shuffle: Poke, poke, and poke. Also, Charm wins games. Land one, watch as your team swarms him like piranhas, and proceed to win 5v4.

Team fights. Your goal is to take out the most dangerous target you can reach without putting yourself at huge risk, and by huge risk, I mean getting instagibbed or locked down by chain CC. Charm said target, unload everything and spam immediately as they come off on cooldown. Aim for the squishies if you can. Reposition or nuke with Spirit Rush as needed. POSITIONING IS KEY.

　
　

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This gets its own section since so many people argue on this point, so these are my thoughts.

Early sustain
Catalyst of Aeons admittedly has the advantage here. However, safe play and passive also provides good sustain. I would normally say it's situational here, but the next point is why I argue against Rod of Ages.

Items fully built
Ahri does not have serious mana problems. Therefore, the mana on Rod of Ages is a nice bonus, but not necessary. Both items give the same amount of AP. Rod of Ages gives a little bit more health than Rylai's Crystal Scepter, but I believe the slow from Rylai's is worth well more than the health and mana bonuses on Rod of Ages. While you're trying to escape with your bonus health or chase someone down, I'll be kiting with foxfire and effectively guaranteeing an escape, as well as securing kills from the absurd slow.

So why not both? If you get RoA first, you delay Rylai's too long. If you get Rylai's first, then you get the stats on RoA too late. Otherwise, uh ... sure, if you think you need all that tankiness, but I feel there are other items with higher priority (deathcap, WotA, CDR).