Many people believe that Ahri is an AP Carry, but I thought differently. I play Ahri as a Mage Assassin. I realized her potential as an Assassin such as Katarina was fantastic and decided to try building her as such, this was my result. This is my first build, so please give me some constructive criticism to help me make it better.

Marks:
Greater Mark of Magic Penetration x 9
Magic Penetration is pretty standard on an AP champion. Alternatively, Greater Mark of Scaling Ability Power can be used for more AP.

Seals:
Greater Seal of Replenishment x 9
Ahri tends to be very mana heavy early game, these runes help. However Greater Seal of Scaling Mana Regeneration are a fantastic alternative.

Glyphs:
Greater Glyph of Cooldown Reduction x 9
Ahri's ganks are very cooldown based, another great alternative for these runes are Greater Glyph of Magic Resist for some magic resist against another AP.

Quintessence:
Greater Quintessence of Magic Penetration x9
Again, Magic Penetration is necessary for an AP Champion, Greater Quintessence of Scaling Ability Power can replace these as well.

Orb of Deception is Ahri's main damaging ability. Max this by lv 9.

Fox-Fire is one of Ahri's best abilities in terms of spell vamp, put a point at lv 4 and max it last.

Charm is Ahri's nuke ability, you want to max this after Orb of Deception.

Spirit Rush is Ahri's ultimate, This is what makes her ganks so perfect, take a point every chance you can.

Essenceu Theftu is Ahri's passive, you want to time your ganks based on this passive.

Highly Recommended:
Ignite allows Ahri to make sure she gets the kill.

Ghost my personal favorite Summoner Spell, great escape skill, good for ganks, good closer, can catch people even after they Flash.

Alternative:
Flash if you don't like Ghost then replace it with Flash since you need an escape skill.

Exhaust can replace Ignite since it becomes more useful in late game, while Ignite dies off around mid game.

Teleport is useful to position yourself to defend a lane, or gank and return to your lane quickly.

Cleanse can be used if the enemy team has heavy CC.

Early Game:
Ahri has some of the best ganks in the game if you play her correctly. I like to start off with an Amplifying Tome and a single Health Potion. If I am doing really well in my lane, already have a couple kills, I'll buy turn that Amplifying Tome into Mejai's Soulstealer, however, if I'm not doing to well, I'll turn it into the Hextech Revolver. My next item are Boots. I will then save up for the item I didn't buy, either Mejai's Soulstealer or Hextech Revolver. You then want to finish your Sorcerer's Shoes.

Mid Game:
At this point in the game, I ask "How am I doing?" If you are doing really well, go ahead and buy Will of the Ancients. If you're doing ok, but are having heavy mana problems, buy Tear of the Goddess. If you're doing poorly, or need more survivability, Giant's Belt is the way to go. After I have these three items, I buy a Blasting Wand.

End Game:
Now everyone is starting to hit lv 16, turn your Blasting Wand into either Archangel's Staff or Rylai's Crystal Scepter and finish off with your Rabadon's Deathcap and don't forget to buy your Elixir's when you have the money.

This is my build for Ahri. Once again, this is my first build so I'd really appreciate some constructive criticism, I'd appreciate reasoning for a downvote, so please comment.