Strengths:

[*] Extremely Versatile; Is known as a "jack of all trades". Has everthing a mid laner needs (damage, mobility, sustain, cc, waveclear) and has no true counters. Can be built and played with a variety of different builds and playstyles.
[*] High kill/assist potential from E; The power of Ahri's charm alone makes her relevant even when behind and helpful to any team.
[*] Power spikes hard after first item
[*] Easy to learn, hard to master
[*] When used correctly, mobility makes her more forgiving than other mages. She can get away with overagressive or bad positioning that other mages would be punished for.
[*] Can always be used as a blind pick / first pick.
[*] Gorgeous and has lots of pretty skins. One of Riot's postergirls.

Weaknesses:
[*] Doesn't scale super well. Gets outscaled by alot of other mages (Although I would argue her scaling is alot better since her Q AP ratio got buffed to 100%. Also, building Liandry's on her, which has % Max Health damage, makes her scale alot better).
[*] Difficult to play into teams with alot of CC.
[*] If used poorly, high mobility can be a liability that can cause excessive deaths.
[*] Popularity means high pick/ban rate.
[*] "Jack of all trades, Master of none" Ahri isn't especially good at any one thing. Although she has no counters, she doesn't hard counter anyone else either.

Ahri is very well rounded and always good, and she pays for that by having poor scaling and no true specialty. In my experience, her best matchups are into short - mid range champs with low mobility and her worst ones are against mages who have one or more of the following: high range, high cc, high scaling (i.e. Lux, Veigar, Aurelion Sol, Malzahar). Akshan and Sylas in particular are also rough.

The idea of this build is a callback to the old mythic Liandry's. I believe burn is great on ahri because she has low cooldowns and multiple auto aim abilities than can easily reapply burn damage. This build lends itself to a more poke oriented playstyle, where Ahri can whittle enemies down from afar while using her movement speed to dodge enemy abilities and close in on low health targets. After completing 2nd and 3rd item, you can effortlessly apply crazy burn to enemies just by chasing them with your R and spamming W.

Early Game: Mostly just farm. Ahri isn't anything special pre 6. No shade but shade. Once you hit 6 you can play much more liberally. I generally like to sit in lane as much as possible unless there is a great roam opportunity or my opponent is on the move.

Mid Game: Ahri spikes hardest here and with this build the burn from Blackfire Torch will give you great waveclear and sustained damage. You are strong on your own but better with your team and you can play the role of long rage cc mage in teamfights very well.

Late Game: If you are ahead you can deck squishys and wear down tanks with this build. Your carry potential is high. If behind, play with your team and focus on landing clutch charms.