This is a mana based guide, and since ahri runs out of mana ALOT early game, this build has made me really powerful.

I prefer the usage ofgives you a DoT and a 4th getaway-rush opening. If your Ulti isn't enough to run from/chase someone, it gives you an extra hand.

It's a pretty simple build for both, masteries must change if you choose NOT to use Flash/Ignite combo.

Since there's an early problem with mana, AHRILLY big problem,Archangel is there to change it. Mana+AP.

Then, you get boots. Dem boots be trippin. Boots of Mobility are your number one option in traveling around ganking/helping/farming/whatever.

Rod of Ages gives you a large HP and Mana(!) boost, increasing your AP by DAMN-A-LOT.

Deathcap. Large AP boost. Needisaymore?

Will of the Ancients gives you both AP boost, Spell vamp, and don't forget that it gives spell vamp to your allies as well! Really useful.

Abyssal Scepter. Purely made to decrease champion Magic Resistance and give YOU magic resistance. Switch with Guardian's Angel in case of AD champions. It's gonna make you survive longer and give you time to kill more people.

Don't bother looking at the stats above. With Ahri and that build, you get over 700 AP (I have never exactly completed the build to screenshot the stats, since games end about the time you hit your 5th item)

Farming is done with Q and W, making sure to hit as many creeps as possible with Q.
Always start your passive's counter with Fox-Fire, so that you will be able to quickly use your Ball to steal a large amount of health from the creeps in case you were damaged.

Pushing a little can be bad early game. Try to force the enemy champion in your lane to go back to heal, use pots and make sure to waste his time as you rack up minion kills.

What's dangerous, is your mana. Keep an eye at it, and don't be afraid to use a few pots.
By the time you have Archangel's, your mana is good to go, and you're getting loads of AP thanks to it.

When close to the enemy, always start with charm. It charms them, pulling them to you, allowing you to throw your Q and get a double hit on the enemy. Thanks to your Q's True Damage, he should be droppin like a fly.

Your ultimate is just a stronger Fox-Fire combined with Flash. You can still use Fox-Fire with your Ulti on, so take advantage of it.

Always make sure that your ball hits as many enemies as possible, as it will rack you up with your precious spell vampirism stacks.

When running away, try using charm to slow the enemy, and use your Q to inflict loads of damage on the run. Don't forget that Fox-Fire can be activated while on the move.

When chasing, your Ulti and flash are the things you need. The use of E can be really useful if the enemy is faster than you, or he's about to run away.

When picking enemies, always start with Charm, and continue with Q, W, and MAYBE your Ulti if you're sure you're gonna get a kill with it.

Pros of this build

-Early mana problems solved
-Heavy AP and MP-Vamp combo
-Extremely powerful against no-MR targets

Cons
-Really squishy at the beginning
-Takes some time to get a full set of items

That's it for now, comment, suggest and vote.
Thanks for paying attention to my humble guide :D. Hope this helps you out.