Hi, my name isn't relevant for this guide and I main Kassadin since season 6. My main elo is Diamond so don't expect this guide to help you get Challenger or whatever, I'm still learning stuff everyday :) but I hope it can be useful, even for higher ranked players who might still learn a thing or two.

I play Kassadin both midlane and toplane, but with different builds. When I play mid I build pretty standard items, with a preference for Luden's Echo as it synergizes so well with R mana scaling. For the runes I prefer Electrocute over Fleet Footwork but both are good.
My toplane Kassadin is where it gets pretty wild. The idea is to build tank with a bunch of mana, ie Iceborn Gauntlet, Abyssal Mask and Frozen Heart. With this build you don't get damage from AP but from mana, and your primary source of damage will be your R. But this alone won't be enough, and you'll end up using extra ults for damage which you won't have to reposition or disengage. On top of that, your ults cost A LOT of mana, and that's where the W max comes into play. For those of you who don't know, W restores up to 40% missing mana, which can be as high as 2000 on a full build Kassadin. So basically you max W, build Muramana, and play around your insane ult damage paired with auto damage with W and Muramana. I know it sounds really weird, but trust me it's really fun to play.

Anyway let's get straight into it.

Kassadin is an AP melee assassin, with extremely high mobility and damage. He is a hypercarry and outscales most champions in late game; but it takes time for him to scale and he is very weak early on.

Pros:

• Mobility: With an ult litteraly every second at level 16, Kassadin is extremely mobile, to the point of basically rendering flash useless on his targets. His passive is also really useful, allowing him to never get creep-blocked.
  
• Damage: Like all assassins, Kassadin has a good oneshot potential on the ennemy carries, and can be a serious threat to tanks as well. What makes him stand out though is that most of his damage is AoE, meaning he can destroy entire teams on his own if the right circumstances are met.
  
• Survivability: With his passive and Q shield, Kassadin is really good against AP champions. It helps him survive the laning phase against most ranged mages, and his high mobility makes him hard to kill post-6.
  
• Itemization: Kassadin scales on both AP and mana, which allows him to get more value from items. It also gives him more item choices, hence the 2 completely different builds.
  
• Split push: Kassadin is really good as a split pusher; his R and E allow him to quickly push the wave and his W is really strong on towers.

Cons:

• Early game: Kassadin's main weakness is his early game, which is mediocre at best; he has very little damage and chase potential, and no escape. His passive and Q help him survive the laning phase, but are useless against AD matchups, and he will still have to back multiple times and concede a lot of CS when facing poke champions.
  
• Sustain: Kassadin has no sustain at all, which is why players usually take Fleet Footwork, which gives him the sustain he needs throughout the game but nerfs his damage output, burst potential and R cdr. Which rune you take ultimately depends on what you value most between sustain and more damage/mobility.

Honestly just take Flash and Teleport, they are ideal for pretty much every scenario.
If you're unsecure you might want to take Cleanse against a Lissandra for exemple, but you'll take much longer to scale without TP, and your split pushing power will be affected. It's up to you.

[*] I didnt include Fleet Footwork because I rarely play it, but of course you can use that instead of Electrocute, it's up to you. Just don't forget to take Domination secondary with Sudden Impact and Ultimate Hunter. For the rest of the Precision path it's usually best to take Triumph, Legend: Tenacity and Coup de Grace.

AP Kassadin:

[*] Rush Rod of Ages, the sooner you have it the sooner you will be able to roam and get kills. Usually a good timing is around the 10 minute mark, but up to 15 minutes you're still fine. After 15 minutes you will have a hard time scaling. A good alternative is Abyssal Mask, as it doesn't require 10 minutes to stack up and is really good against AP champions. Keep in mind that it doesn't give AP though, even if the passive damage buff kinda makes up for that.

[*] Buy Sheen when you can after having completed your Rod of Ages, it gives good value for money, helps in trading and farming, and you can upgrade it later to Lich Bane or Iceborn Gauntlet depending on how the game goes.

[*] Since the W change you don't have to buy Tear of the Goddess as early as before. Feel free to buy Sheen and even Lost Chapter first.

[*] Luden's Echo is an excellent item on Kassadin, the huge amounts of mana and AP are really good and the passive is nice as well as it adds damage to your burst.

[*] I see a lot of Kassadins go Zhonya's even against AP; in my opinion it's one of the worst items you can build on him. Sure the active is nice, but the cooldown is high, and the item stats are garbage. You might as well just build Iceborn Gauntlet and get more armor, more cdr, a slow and as much damage value. Only buy it if you're against Zed and feeling very insecure.

[*] Some players decide to not build Seraph's Embrace, mainly because it takes time to scale, but since the W change it's not that big of a problem anymore, and the item is too good not to build:
Seraph's Embrace gives 50 Ap, 1400 mana, 20% cdr, refunds 25% of mana spent, gives bonus AP equal to 3% of your maximum mana (which, full build, you have around 5000 of, which gives 150 AP) and on top of that gives you a shield (900 damage shield at max mana). It is THE most valuable item on Kassadin, it just gives so much stats, passives, and an active.

[*] Void Staff and Morellonomicon: I personally prefer the latter as it gives more stats, and the anti-heal is interesting, but of course if the ennemy team is stacking mr just go for Void Staff (or even both).

Tank Kassadin:

[*] Abyssal Mask and Iceborn Gauntlet are excellent items on Kassadin, you should build them every game.

[*] Get Manamune right after you first item if you're ahead, if not just build Tear and upgrade it later.

[*] Righteous Glory is amazing to catch carries; with your ults it is practically impossible for them to run away.

[*] If the ennemy team is not primarily AD or auto-attack based, you can build Iceborn/Abyssal/Manamune/Righteous and not go for Frozen Heart but Archangel's Staff instead.

[*] If the ennemy team is full AD and you or your team are really far behind, build Randuin's Omen or Sunfire Cape instead of Manamune. Keep in mind you will have significantly less mana and damage though.

AP Kassadin:

As a late game hypercarry, your goal is to farm, farm and farm until you reach your lvl 16 powerspike. Play it safe early, concede CS if you have to, and don't do anything too fancy.

Early game:

Against ranged opponents, pre-6 Kassadin is extremely weak, but when you hit level 6 you can relieve some of the pressure and start looking for trades and/or all-ins. Your ult also gives you the possibility to roam easily and try to grab a few kills botlane. Against melee opponents it's a bit easier as you can use your W if they get in close range to attack you.
In most matchups you will end up having to last hit under tower though. Just farm safely using your abilities until level 6 as Kassadin has a pretty good kit for under tower last hitting.

If your jungler comes to gank before level 3, unless the ennemy is really overextended just ping them back. Without his E, Kassadin has 0 follow-up potential and very little damage this early into the game.

During the laning phase, spam your E whenever it's up; use it to last hit from a safe distance, to poke your opponent, or to slow him in order to prevent an all-in. If you get low on mana, keep in mind that your W restores 5 times more mana when used on champions, but don't put yourself in a dangerous situation just for some mana. If you're under pressure, just use it on minions whenever it's up.

Once you hit 6, a good trading combo is waiting for 5 E stacks and going R + E and then back away; the E slow will usually prevent them from trading back. Only do that if you know you aren't getting ganked, as you are using your only escape (R cooldown is quite high at rank 1).

Most of the time you won't be able to all-in before levels 9-11 unless your opponent makes a huge mistake. Wait for powerspikes, ie Rod of Ages, level 9 (E max) or level 11 (rank 2 ult) to surprise your opponent and go for the kill.

Roam whenever you can, botlane being ideal but you can also roam top if the ennemy toplaner is overextended. Use your R to go through walls and stay out of vision.

Once laning phase is over, you want to split and get as much gold and xp as you can, the goal is to get to level 16 as fast as possible, while keeping an eye on the minimap to be able to TP in when your team needs you.

Late game:

In late game, your goal as an assassin is to get to the carries and one shot them. Kassadin's extremely high mobility and his ability to go through walls allows him to flank the ennemy backline very easily. Just wait for your team to engage the teamfight and riftwalk from behind to surprise and kill the ennemy carries.
You can also try to catch isolated targets in the jungle, one shot them and riftwalk away to regroup with your team and go for objectives.
Kassadin is also a really good late game splitpusher as he has decent waveclear, deals heavy damage to turrets and can 1v1 almost any champion at full build. If you have TP up, split bot/top and tell your team to pressure Baron/Elder.

Tank Kassadin:

With this build you'll pretty much be the same late game monster, but you're sacrificing a bit of damage and mobility for some extra resistance.

Early game:

The early game will be much easier than with AP Kassadin, especially vs melee. You can out-trade almost any champion level 1 with your W if you stay near your caster minions.
Against ranged or high poke champions such as Pantheon, play as if you were mid, and keep in mind that sometimes the gold you get from a minion is not worth the damage you will take for it. It's ok if you miss a few CS, you will outscale everyone anyway.

Don't forget to ward a lot, as a pre-6 Kassadin top will attract more than one ennemy jungler on your toplane.

When trading, remember to put in as much autos as possible as we are maxing W, and whenever the active is up try to AA reset.

You can really become a lane bully once you get level 6; the most efficient combo being stacking Grasp on minions, and then going for R + W before backing off. Do that again and again, until either they get too low on HP and are forced to back (if they don't just kill them with your stacked ult), or they still have enough HP, in which case just wait for your ult stacks to reset and start again.

Late game:

If you are a bit behind, you should split push and continue farming until you hit your level 16 powerspike.
If you're ahead you can choose to split push, as this will force the ennemy team to send at least 2 players to stop you, which gives your team the teamfight advantage, or you can group with your team and take objectives or force teamfights.
With this build Kassadin is really hard to kill and deals obscene amounts of damage; in teamfights you can go for the carries or stay in the frontlane, tanking for your team and damaging the ennemy tanks.
Before engaging, wait for the ennemy team to use their main CCs on your team and then go in and one shot everyone.

Vision in the ennemy jungle is extremely important as spotting isolated targets will allow you to quickly kill them and regroup for the 5v4 advantage.

Post-16 you can really abuse your ultimate as it allows you to alternate quickly between split pushing, catching ennemies and grouping.

[*] You can Flash during your ult animation to double the distance and surprise your target. To do that just press your Flash key right after you use your ult.

[*] Similarly, you can flash your E animation, which makes it much quicker than Flash + E, and gives less reaction time to your target.

[*] Your Q cancels Malzahar's ult. If you time it properly, this can be game-changing. It also cancels TP, and one fun thing to do if you're facing a laner with TP is to Q them right before they back, so that it doesn't cancel their back but instead follows them to their base. If they TP right away to get back to lane they will be quite surprised :p

[*] Your W resets your auto-attack timer, try to use that when you can.

[*] What most players don't know is that Kassadin's W extends his auto attack range, which can be useful for trading and in some situations. For instance, if you're chasing a low HP ennemy just outside your normal attack range and don't have any mana left, it can save you a flash.

[*] If you're doing Drake or Baron, ask your team to spam their abilities. This will charge up your E and you'll be able to spam it as the ability has a pretty low cooldown.

[*] Q's travel speed is pretty slow, but the shield activates before the animation, not when the Q actually hits, so keep that in mind when trying to block damage.

[*] Your ult stacks every time you use it, and the damage goes up with each stack. If you just backed or died you can use it 2 or 3 times to get back to lane (depending on your mana), and instantly go for a trade with your stacked ult; the damage will surprise your opponent.

[*] When chasing someone, especially in the jungle, look for walls to extend your ult range. Like with all blinks, if you click passed the middle of the wall, you'll end up on the other side.