This build focuses on Ahri's strengths. It quickly caps cooldown reduction, and then boosts survivability. At first, this may seem like a tank build, but with Ahri's spells not scaling very well with AP, (roughly .35 each ability), it is wisest to put out as many hits as possible and survive in order to do the maximum damage to the opponent.

The reason Rylai's is built so quickly is to aid in both pursuit and retreat. The slow means that if you are being chased, you can pop fox fire in order to slow the opponent, and then fire charm in order to completely eliminate the threat and escape. While ganking or going in for the kill, spirit rush coupled with charm make the foe easy to pin down. Frozen heart helps to cap CDR, and also adds some survivability against AD champions such as yi or Vayne. If you are having trouble with AP opponents, I recommend going for Banshee's Veil first. The Will of the Ancients also provides a boost to survivability late game, as the spell vamp will benefit both you and your team. Lastly, the Guardian Angel is purchased to provide more sustained pushes and prevent a wipe from completely losing the match.

As far as the summoner's spells go, the two i prefer are more geared toward pursuit or escape, but ignite is definitely a good choice as well. Whatever floats your boat in this case.

All in all, not the most average build for a mage assassin, but Ahri is not a normal mage champion. Her scaling with AP does not provide the same DPS burst and huge hits that others like Ryze can put out. The major pros to Ahri are her survivability, her charm, and her ultimate. It is best to play her as a sniper at a safe distance, charming enemies that are careless, and when someone is low, you can ult in and finish them off, having enough survivability to get back out.

Pros: Provide a steady stream of DPS while avoiding death. Does extremely well early game, and in team fights.

Cons: The major downside is a lack of "bursty" huge hits. This is countered by a large number of spells in a short period of time. Also, late game can be tricky, as other AP or AD champions will be buffing burst, you will be focusing on survivability and utility.