The summoner spells for ahri are pretty intuitive, you will always want flash and 99% of the games ignite.
Due to ahri's lack of burst damage in the early game ignite is essential to make sure you can get those early kills to help you go into the mid game strong.
In the other 1% of the games, I recommend Cleanse, but you will only want this summoner spell against team comps with a lot of undodgeable CC, such as Twisted fate, Fiddlesticks, lissandra, etc..

These are all the items you want to buy on Ahri.
The question is when to buy them. (always buy luden's echo, deathcap and void)

Protobelt: Buy if you get an early lead. This item is very cheap and efficient and once you complete it it's a huge powerspike, you should be able to kill squishies without even needing to use your charm and you can sometimes use it to reposition yourself to make sure you land Q or even dodge a skillshot or something. It's the best item for ahri if you're ahead and if you manage to keep this lead, consider a mejais.
After that you usually just want to get deathcap and void. The order of these 2 depends on how much magic resistance the enemy team has, if everyone is rushing MR, go void first, if not, just go for the deathcap.

Banshees: Usually you want ths item if the enemy team has a lot of AP or if you feel like you need to dodge some sort of engage CC, such as nautilus, leona, etc. It's a good defensive item, but it's not optimal if you want to make sure you're able to 1 shot people.

Zhonyas: You want this vs teams who are either full AD or have some sort of really annoying AD Asassin that you cannot outplay using your ulti.
I personally don't like this item on ahri, your 3 ultimates are more than enough for you to be able to run away from a zed or something, but in some cases they're just so strong you don't have enough time to react, that's when you should buy this item, specially if you're struggling in lane against them.

Morelo: A lot of people buy this item 2nd on ahri, I do not recommend it. It is not a good item, it's very expensive and doesn't give you great stats. Only buy this against teams with a lot of healing (yuumi, soraka, vlad, etc) and if absolutely necessary. Your job is to try to ***sasinate someone in the fight, not to be a utility mage.

Liandry: This item is incredbly underrated on ahri, you should look to build it against teams with a lot of bruisers/tanks. After you have deathcap/void it doesn't really matter what you build since you should be able to one shot squishies, however if the enemy team has a huge frontline you can buy this item to deal with them and play as a combat mage.

I think the only thing some people could disagree here is W 2nd and W max instead of E max after Q.
Why do I go W as the second skill?
It's simple, if you go E you'll have very little trade potential in lane and you simply will have to be zoned out of cs by the enemy because in higher elos they're just gonna sit behind creeps so that you can't charm, by going W second, you're able to easily proc eletrocute and possibly get a good trade in laning phase.

As for the W max instead of E, maxing E only gives you a tiny bit of extra dmaage and CC, it's not relevant for ahri's playstyle which should be like an assassin, maxing W gives you more damage and lower cooldown which is really helpful in long extended fights and a lot better to assasinate squishies.

These are the standard ahri runes for me.
Eletrocute is just standard for ahri since she plays like an assassin, you could play glacial but this guide is focused on eletrocute.
Taste of blood here is very debatable, ahri can use all 3 runes, but I strongly suggest ToB to make sure you have the best laning phase possible. Ahri is an outdatetd champion so you need every help you can get to make sure you don't fall behind during lane phase, since you don't really scale that well, this rune is not optimal for damage but it will help you in lane and maybe you can create small advanatges out of that.
Eyeball collection is just the easiest one to stack out of all 3, no secret here.
Ultimate Hunter is the best one out of this tree on ahri, she is 100% reliable on her ulti to make plays and having acess to it more often only brings advanatges to you.

Biscuit and Time Warp:

This is where most people would probably disagree with me, but like I said earlier you need to have a strong laning phase on ahri and these are the 2 runes that will make sure you do.
By taking these runes you can play laning phase more agressivly and go for more trades more often, you'll have mana and health sustain from corrupting potion and also a buff on movement speed which is really helpful during the entire early/mid game as it can help you rotate the map and maybe even dodge skillshots or avoid ganks.

I made the threat list based on how difficult the laning phase against the champions are, I'm not taking into consideration teamfights or anything, there are champions who are very easy to lane against such as malzahar but he can completely destroy you in a teamfight just by pressing R. So this list will be focused exclusevely on the matchups and how to play them.

[*] Level 5 Threats - Unwinnable matchups

[*] Akali - Ever since she got mini reworked I find this matchup to be unwinnable for ahri, you can try to win some trades before level 6 but after that there's no going back for you. Try to poke as much as you can level 1-2, make sure you never fight in her shroud, and save your charm if she decides to go in. After level 6 try not to get in range for her ulti, she can use her 2nd dash faster than you can get away with your 3, so holding up to your charm is your most valuable tool to play this matchup and against every champion with a dash in general, since it will likely prevent them from going all in at you.

[*] Fizz - It's basically the same as akali, the difference is this lane is absolutely over for you after level 3, even trying to poke him is risky since his combo is so easy to land and can easily take half of your hp away, so if you do it more than once, you're just gonna die and it's over for you. If you do manage to survive until 6 without dying, you just have to save your R for when he uses his fish. Forcing a fight here is still not optimatal even if you have ulti advantage, you have to rely on him taking a lot of free damage to then all in.

[*] Diana - The fizz and akali matchup are very similar, since they're both assassins with a lot of mobility and a bad level 1/2, the diana matchup can be a lot more annoying because you unlike fizz and akali, she can trade back your poke at level 1 and at level 2 she has a shield that can deny your damage. Which means that you already enter lane losing in this matchup. You should really never be able to win this one, specially if she goes RoA first item, she'll just be too tanky and everytime she goes in she probably just forces you to ulti away, so just save your charm and do your best to interrupt her dash. If you force trades, make sure you try to do it when she can't shield and you have eletrocute up.

[*] Sylas - I would say this matchup is a bit more similar to diana, but it works in a different way. The reason why you don't win this is because sylas is really good at extended fights where he can proc his passive auto many times, heal back and abuse conqueror. However if you're really good at dodging skilshots you can win trades in this matchup until he gets level 6 and his first back. That's your only window to win it. After that it's pretty much over since he can use your own ulti better than you and don't forget he has a dash of his own aswell, so you can't outrun him.

I'll finish it later :c