Deathfire Grasp: As stated above, this item is a really good item for Ahri because it provides Ahri with her most needed stats: AP and CDR and also helps her burst down squishy enemies like, for example, the adc of the enemy team.
Sorcerer's Shoes: Provides Ahri with a good amount of magic penetration and increases her damage.
Abyssal Scepter: Now this item is an item you will absolutely love on Ahri because, simply, it reduces the magic resistance of nearby enemies by 20. Combined with your ult ( for gap closing ) will make you look like a huge threat to enemy team and especially enemy adc and apc because you will blow them up in one combo. Also, this item will help you, with its magic resistance stat, if you are facing an ap champion.
Rabadon's Deathcup: a HUGE spike to your ap stat because not only it gives +120 ap but also 25% more ap of your current ap.
Void Staff: This item provides you with more magic penetration and combined with abyssal scepter, your magic penetration runes and your sorcerer's shoes will help you deal with tankier enemies. Also, it adds a nice buff to your ap stat.
Zhonyas Hourglass: in some cases you might think that its a core item for Ahri, in some cases you won't really need it. Anyway, I suggest it because it adds a lovely buff to your ap stat, to your armor (really useful if you are facing an ad champion) and its active use will help you avoid unnecessary damage and cc abilities (great example of its use is zed's deathmark).
IMPORTANT: If you are facing a hard cc champions at lane (like veigar etc.) you will probably have to go with mercury treads (tenacity= cc reduction). Penetration from runes only is probably enough for the time being to deal with them.
IMPORTANT 2: if you have trouble laning against ad champions and you are behind in lane (trying to farm under turret with the fear of getting dived and dying in a matter of seconds) you probably have to pick zhonyas hourglass as your first item.
Deathfire Grasp is really useful early game as it gives Ahri her most needed stats (cdr and flat ap) and with its active you can burst someone down for an easy kill. Moving on to Sorc shoes, all of you know that its the core boots for ap champions (especially to champions with cdr items like Ahri with DFG). I dont like repeating myself but I advice you to never go cdr boots, Sorcerer's Shoes are the best boots for Ahri. Abyssal Scepter will help you reduce an enemy champion's magic resistance by 20,therefore it will help you kill enemies easier (example: enemy adc has 40 magic resist? with that item he will have only 20 so its gonna be easier to kill). Rabadon's Deathcup adds a lot of much needed ap to your stats. Void Staff provides you with a lot of penetration to help you kill tankier enemies. Zhonyas Hourglass is also a pretty nice boost to your ap stat and also protects you from a lot of cc and damage especially if you are the primary target of the enemy team and you have to get focused in fights. Also helps a lot, if you are laning against ad champions mid, like talon,riven and zed with either a big burst or decent poke.