Building Ahri

You want to get Ahri a lot of AP so go for a lot of the AP items such as Deathfire Grasp and Rabadon's Cap. I recommend the boot start and 2 health potions and 2 mana potions. Get the sorcerer's shoes and then start building to rabadon's cap. After Rabadon's cap go ahead and start going for a defensive item. A good defensive item for Ahri is Zohnya's Hourglass or Rylai's Crystal Scepter. In the end you should have your final build as sorcerer's shoes, Rabadon's Cap, Rylai's Crystal Scepter, Zohnya's Hourglass, Liandry's Torment and Deathfire Grasp

I agree with Lyoko somewhat, though I hate when you only get one possible build as an answer for these things because in 2000+ games of LoL, not one of them has ever been exactly the same. You can find a few items that each work on the given champion in a given situation, but there's no magical 6 item set that will be optimal in EVERY situation, so...here's this.

Starting Items:

Boots + 4 pots: A reasonable start if you're going against a mid-laner who relies heavily on skillshots (ex. Brand/Ezreal) and a jungler with extremely limited gank potential pre-6 (ex. Warwick/Amumu). Only problem with this is that you have minimal power and only have a set time you can stay in lane and only a set amount of harass you can put out before having to retreat.

Doran's Ring + 2 pots (typically 1 hp and 1 mana but can be 2 hp): A good start for when you KNOW you have an early advantage and want to exploit it for all it's worth before the other guy hits lvl 3 or 4 and can respond (ex. Fizz). Gives you more raw power and slightly better mana sustain (and for Ahri, mana sustain = hp sustain) but again, be wary of ganks.

Faerie Charm + 6 pots + 1 ward: Safest route. The charm gives good mana sustain and builds into several items Ahri would find useful. The pots still give you a set time to stay in lane, but an extra 2 pots can go a loooong way. The ward is most important for obvious reasons. This is a good start for when you're playing against a tough lane (Ex. Kass/Zed) or one where you have you have minimal kill potential in lane and need to farm it out until you can gank side lanes (Morg/Galio).

The boots should go into Sorc's 99% of the time, but don't think Tabi, Mercs, and Mobility are blocked from you. If for some reason you're going against a 5 adc team, get Tabi. Period.

Offensive Options (Emphasis on the "Options"

Deathfire Grasp: The core item for any and every AP Burst Mage. If you can remember to use it's active before attacking someone, they'll melt like a lightsaber through warm butter. This should be your first complete item ~80% of the time

Rabadon's Deathcap: Another core item for any heavy AP champion.

Void Staff: If you're doing your job right, they'll start building a good bit of MR. This will fix that.

Morellonomicon: Remember that Faerie Charm from way back? Here's one way it could work out. Good AP, mana regen, and the 20% CDR means you can actually afford to let your jungler keep their blue buff (heresy, I know). Plus if that Mundo decides to get smart and ult as you come from behind and start bursting him down, he's still going to die because of the passive. A rather underrated item, imo.

Liandry's Torment: Not the best fit for a burst-mage, but if they're so big you can't burst them all the way down in 1 cycle, this will help get them down just a bit lower for your spells to come off CD and finish them.

Will of the Ancients: Rare buy, but sometimes it's worth it.

(Defense on next post)

Thx ill try it out as soon as possible!

Ps... Nice names

for ahri, I like going for the mana chalice due to my constant spamming of spells. If I am doing well in lane, I will go for a dcap if not doing so hot I will finish out athene's first. After that depends on who you are up against. My typical end game build: sorc boots, dcap, athenes, void staff, rylia's, and zhonya's/ga.