Pros
Ahri is a strong mage
Does alot of damage early game
Clears out Minion waves quickly
Is a badass
Eats Peoples souls

Cons
Early Mana Problems
Cant take out turrets super fast
Is kinda squishy early game
Eatting peoples souls is sadly a pro and a con

Since Deathfire Grasp Grasp was taken out I find it best if you get Deathcap first because of the HUGE AP boost,and even though people get boots first I think youll do better it you get this first.
Getting Sorcerers's Shoes gives you magic penetration which is great for mages and bursty champs
Rod of Ages might seem weird on a buster like Ahri but ive find it helpful it makes your mana problems disappear as well as your heath I know you could just get the Morellonomicon but Rod of Ages gives more AP per stack which is why it works better!
Next is the Lunden's Echo "the new Deathfire Grasp", It's great late game because it game hit up to four enemies at once doing 100+ magic damage it might be short range but I find it can be very usefull
Zhonay's Hourglass should be the last item you get unless you're losing by alot then get it before Lunden's Echo, Other wise the AP boost helps clear out enemies fast and if you get in a sticky situation you can use it while you wait for your team to help you.
Void Staff is an item you should get if you're kicking everyone's ***, If you aren't and your up against an AP player get Abyssal Scepter to counter them. If you're up against a AD player get Thornmail or Rylai's Crystal Scepter. Thornmail will reduce the damage they dude but Rylai's Crystal Scepter will give you more healt and AP, your choice just depends on the game.

Ahri is a wonder champion, So help other lanes if they need it and you're lane is okay. While in teamfights cast your charm to help your team get easy kills.
When pushing mid it can be hard to tell if you should leave to help your team of stay. If the enemy in your lane is missing keep pushing until you can see them on the map being careful and placing your ward in a near by bush, if the enemy is discovered in a different lane make your way there to help.

Orb of Deception is good for helping you clear minion waves it is also a very powerful poke.

Fox-Fire is used best ride after you hit your charm or right before you start your ult

Charm is great for pulling weak enemies out of crowns or to stop strong ones as you get away

Spirit Rush Lets you dash up to 3 times and hit your enemy with three attacks as you're doing it, it can be used to catch up to enemies or to run away from them. Try not to get ahead of you're team though you might end up in trouble

Your Passive Essence Theft is why you don't need two health potions at the start of the game. When you use it try and hit as many things as possible to regain as much heath as you can.