Ahri in my opinion is one of the best AP mids in the game. She has the ability to snowball a lane and effect the whole game using her high damage and mobility. Ahri is my most played champion mainly because of how fun she is to play. When playing Ahri you must understand that she is an extremely high skill cap champion, excluding the ability to land skill shots the main difference between good Ahri players and bad ones is the use of the ultimate 'spirit rush'. This can be used offensively but this requires caution. Whilst playing Ahri like any AP carry cc and silences are you main weakness, many unskilled ahri players will use the ultimate to initate a fight but this puts you at risk, Ahri's skills give a good combination of high burst and sustained damage, never put your self in a high risk position just to kill the enemy AD. Pre DFG nerf Ahri had the ability to easily 1 shot the enemy AD but now it is not worth dying. After getting a Rylai's Ahri is one ofthe best AP carrys at kiting. Using the slow from all your spells and the 3 charges on your ultimate the enemy will not be able to catch you, with a blue buff you will easily be able to tear them down.
Ahri is best in the mid game, she gains a huge power spike when you get a deathcap. You must gank your friendly lanes, ganking lanes as a mid champion successfully will win you the game. One thing you must not do when you are in the laning phase after level 6 in stand in mid lane waiting for the next wave of creeps. Either steal your jungle off the jungler or roam. If the enemy mid laner leaves the lane you either follow or push the tower.
The ideal warding around mid lane is having one ward at the enemy wraiths and one in the bush at mid lane on the opposite side. This gives you safety when taking your wraiths and the enemy wraiths and preventing their jungler ganking you.
In a 1v1 situation be sure to use 'spirit rush' intelligently, this can involve jumping over walls or avoiding skill shots, but also the ultimate can be used to land your own skill shots.
Ahri is a great champion to have when small 2v2 fights break out in the jungle, she can jump over walls to get the the situation quickly and has the cc and aoe damage to win the fights.

Ahri's main counters in lane are Kassadin, Fizz, Katarina, Annie and Leblanc. These are mainly champions that can avoid your charm or 100-0 you faster than you can. But a skilled Ahri player can beat all of the by outplaying them with your ultimate. Ahri's easiest lanes are ones where she is under little pressure and can freely roam and bully the enemy champion for example Karthus and ziggs. Ziggs can be a pain before Ahri gains her first blue because his ability you harrass is higher. No one plays Xerath but Ahri destroys him because she can dodge his ulti. Ahri destroys Morgana and out roams her.
Roaming doesn't just mean ganking it also means counter ganking, make sure you have high map awareness and you are looking around the map at all times. If the enemy jungler in ganking bot and they look like they may turret dive thrwo at q at the minions in mid lane and run to bot lane. If the have dived they will all be low, Ahri is amazing at clearing up these situations and they are easy triple kills for you.
I know it seems obvious but if you can get an early advantage as Ahri in lane you can shutdown the opposition mid laner very hard. This can be done by early camping from your jungler. When you hit your taunt it is almost guaranteed kill. After you are 6 and ahead you can ulti and charm your lane for easy kills. When you are farm enough ahead your lane will be unable to follow your when you roam so you can create easy 3v2s.

I take the standard AP Quintessence, Magic pen reds and AP/Level blues. For yelllows there are a few options, health/level, AP/level, mana regen or magic resist. I prefer health/level, mana regen isnt needed after chalice/blue buff.

If you are against a hard lane such as Annie you can go 21/9/0, getting 3 points in Resistance 1 in Hardiness, 4 in Durability and 1 in Veteran's Scars.

Athene's is an amazing item for most mid laners and after the DFG nerf is a must for Ahri. It gives CDR mana regen and magic resist. It enables you to farm without blue and 1v1 your lane.
Blue buff is essential for Ahri as it provides the CDR and mana regen to push the lane and out roam her lane.
Abyssal is in my opinion a less viable Athene's. However it is useful against lanes like Annie or Leblanc that at lvl 6 can 1 burst you. Rushing the magic resist is good. If you go with abyssal you mush get double dorans also, this makes this option expensive. Only get dorans with athene's if you know you wont be getting 2nd blue or if you get fed extremely early for example a first blood on an invade, the early dorans will help win the lane.
Merc Treads always when against veigar.
Possibly QSS when agianst Malz but this depends on the situation, I prefer not to and just to be sure I am never close to him by pushing lane we long range Q and roaming.
Don't buy spell vamp, you don't need it because of the OP passive. Other items are more useful.
Rod of Ages, average item on Ahri. Rylai's is so much better and you only want to get 1 item with health.

Flash + ignite is standard.

Flash + TP could be used against a Twisted Fate if you are worried you will be unable to keep up with his roaming, but i would still go ignite and when he leaves lane push mid hard.

After lvl 7-9ish your q will 1 hit ranged minions. Fire a q and w whilst standing on your side of melee minions to just about 1 hit the wave.
Farming wraiths is easy on Ahri. Fire a q into the camp from mid lane whislt standing next to the wall. When you get vision of the big wraith press w and the w will almost 1 hit it, finish it off with basic attacks, by this time the small wraiths will have run around to you, 1 q will finish these.
You can start to take wraiths after you get blue/chalice. Often if you have their wraiths warded it is advantageous to leave the small ones just stealing the big one.

Kite everyone all day dealing as much damage to high profile targets as possible without putting your self at risk. Use charm to catch people out before the fight a charm on the enemy AD at the start will win the fight. Never ulti into the middle of a fight (with hourglass you can but i value GA higher on Ahri)

Pros:

High mobility
High damage
Safe lane after 6
Insane roaming
Quick pusher
Scales into late game well
Has some light cc

Cons:

Easy gank for enemy pre 6
Weak vs many match ups in mid lane
Hard to team fight well with if you have little experience

To conclude Ahri is a hard champion to master and requires time to understand when it is best to use what skills. If you have the ability to hit skill shots it is worth playing Ahri as she is fun and when fed is OP.
Ahri works well on teams that can use your ability to catch someone out with charm. Shen or Maokai are example of champions that can land free cc on a taunted champion. Maokai is great at ganking an Ahri lane because the charm cant miss if you save if for when the target is rooted.