Ahri, like a lot of ad char, can't handle a lot of damage. That why it s important to focus minions before your ulti. The gameplay is the same as kassadin in a way, you can't kill someone beeing sure that you will kill him before he will kill you.

So focus minion, lvl 6 you're suppose to have your mejai & ur simple boots. Now you can start your hunting (under turret like kassadin again :p)

You have to know exactly your max range, because an ad carry can kill you with 4-5 auto-shot (i speak earlie game). All the time MAX RANGE. That why exhaust might be a good spell for you if you don't really know your exact max range.

The coffee cat ahri is a char who give you the time to drink coffee & roll your cigz coz of this "warmup" early game, after lvl 11 you will be in the position of engaging ennemy easily.

Think about use charm & ulti priority against ad! Never use it to engage a battle, prefer the way to use it in the middle of a teamfight, as a burst middle fight or to escap if the ennemy is pwning you

I m use to put mana/5sec as yellow rune to help with a mejai build.
The rest is kind of logical, i just prefer my blue runes with +x/lvl because it give you advantage after 20mn in the game.

My skill sequence depend on my job.

As mid i will take 1 lvl on each spell than boost my orb & my fox-fire, but 1st orb than fox-fire.
Fox-fire is a good skill but this spells required low range, so i use it as combo with ulti or when i m hunting / running (until lvl 12)

So basicly i play against the ennemy with my orb, & when he is btw 75% & 50% i use ulti once / fox-fire / ulti scd / orb / ulti thd / fox-fire (& ignite just in case somewhere in the sequence if you feel using it)

On lane, i prefer to start with the charm and then upgrade orb & fox-fire at the maximum (combine to xin/cho' or etc... with a stun / bump / snare / ... it give you easy the first kill)

I put ignite as attack spell to help me earlier game & have 6-8 stack before lvl 10 on my skill sequence.
In the other hands, i put ignite as defense spell that i combine with fox-fire, when someone is hunting you, fox-fire -> ignite -> (no break time during casting) -> fox-fire & your ennemy is btw 75% & 50% hp. Just turn you, charm -> orb -> fox-fire -> orb & he is suppose to die

Flash is not mandatory, but, what i did, & will use as example is, when you have ward at baron area, & you want to steal it, you can use your ulti to make damage & combine fox-fire for champ (& ignite or what ever on baron). 3 use on ulti = go at baron / come back jungle / go at baron -> block. You can use your flash easily as a first or X sequence ulti.
Ahri ulti give you the possibilitie to cross 2 turret in 2 sec, an ennemy can't stay 1 turret back as you. So you can flash as offensive, flash -> ulti (may hit ennemy) / fox-fire / ulti / orb / fox-fire.
You will have a /all ***in" OP champ coz you cross half of the map on 6 GCD -> 6 Sec

Exhaust is a good spells to play def. It give you range for ad (combine with charm & ulti they can't touch you) or it give you an easy way to kitting/run if you charm one & exhaust the other one (charm = snare)