I'm a Kassadin main that achieved this season on Eune grandmaster with 92%WR on Kassadin and Challenger with 77%WR on that champion.

-General information about Kassadin
-Tips for champion select
-Runes
-Starting items
-Core items
-Situational items
-Skill maxing order
-Matchups
-Basic mechanics
-Combos
-Your biggest powerspikes
-Tips for laning phase
-Tips for Mid/Late game

Kassadin is a midlane mage/assassin that scales heavily into late game, he is a great champion for 1v5'ing, but he might struggle a lot early game. He has some great matchups on mid, however he also has some awful matchups.

As I already said, Kass has some good and bad matchups that you want to avoid, in order to do that, I wouldn't recommend you blind picking him. If you are picking before enemy midlaner it's better to ask your teammate that picks his champion later than you if you can pick his champion and then you can trade. Also keep in mind that in current meta there is a lot of flex picks that Kassadin struggles with on lane such as: Tristana, Irelia, Kled, Aatrox, Urgot etc.

Kassadin is a champion that can make a use of a lot of different rune pages, but here are 3 best and most common:
https://prnt.sc/pltt8l I use this runepage in hard ranged matchups such as ryze, xerath (tp preferable)
https://prnt.sc/plttmf I use this runepage in most of the melee matchups such as fizz, yasuo (always take ignite with this runepage, because it might be hard sometimes to proc electrocute early and ignite counts as an attack/ability
https://prnt.sc/pltty0 I use this runepage in easy ranged matchups such as vladimir, orianna (both ignite/tp good)
Notes: In your defense rune (the last one) you should take either armor or magic resists depending on your lane opponent, also if you play vs ad champion I recommend you to take double armor since your starting armor number is just 19.
If you take fleet footwork you might not need legend:tenacity if enemies lack CC, in that case attack speed is the right option since it really helps you with last hitting in early laning phase.

Starting item hardly relies on your runepage.
If you took:
Fleet footwork - you wanna start with Dark seal + Refillable potion.
Electrocute - you wanna start with Corrupting potion/Doran's ring + 2 health pots.
Arcane Comet - you wanna start with Doran's ring + 2 health pots since it increases your damage on both Q and Arcane Comet, scorch poke.
Notes:
In very hard matchups you can grab Doran's shield + 1 health pot.

The perfect item path is:
Tear>Catalyst>ROA>Finish Archangel's Staff from you Tear>Lich Bane>Rabadon
But as we all know, it's usually not that easy as above.
If you struggle a lot you should abond early Tear and rush Catalyst into ROA, then Tear and build as usual.
If you struggle even harder vs AD matchups you can rush Armseeker>Catalyst>ROA, then Tear>Archangel's Staff>Zhonya's (if you are still struggling)>Lich Bane>Rabadon

Boots: Don't buy Ionian boots!!! There are only 3 options here: Sorcerers/Tabis/Mercurials

Notes: If you are mega ahead, don't be afraid to buy Rabadon as you 3rd item instead of Lich Bane, also if you have some stacks on your Dark Seal you can think about upgrading it to Mejai's.

In some scenarios, where enemies are heavy AD/AP you can build some different items than usual.
If enemies are heavy AD: Consider buying Zhonya's/Tabis or even Frozen Heart as 4th/5th item. Yes I just said Frozen Heart, this item gives you insane amounts of CDR, Armors and Mana which increases your Ultimate damage. This item also reduces enemy attack speed which is very good vs champions such as Yasuo and all ADCs.
If enemies are heavy AP: Consider buying Abyssal mask instead of ROA, Mercurials and Banshee's that gives you a very important dose of CDR.

There is a huge variety of skill ordering on Kassadin.
Most basic is: Q>W>Q>E>Q>R then max Q>E>W - I use this mostly in Hard-Medium AP matchups.
Most snowballing is: Q>W>E>E>E>R then max E>W>Q - I use this in ad matchups and Easy AP matchups. For this you need an early Lich Bane because this build hardly relies on one shot made by R+W.
You can also consider: Q>W>E>E>E>R and maxing E>Q>W if you need waveclear early, but you're not ahead enough to one shot enemies with R+W.

Notes:
In melee matchups you might take W first and try to trade lvl 1 by resetting you autoattack.
Upgrade R whenever possible.

In early laning phase you should use your shield from Q to block enemy magic damage, you can even use it on minions if you don't have range on enemy champion
Your Q cancels channels, this means you can stop Xerath from channeling his Q or R, Vladimir's E, Fiddlesticks' W and R, Katarina's R or even channeling Herald etc.
Your W is an auto attack reset.
Your E is loaded by both ally and enemy abilities.
You can extend your R range by using R+Flash
You should stack your R as much as possible before a fight, but don't go to low on your mana.
Keep in mind that your W grants you mana back based on your missing mana, that means, in late game when you have 800Mana and you use your fully stacked R and then W a champion you will be able to cast your R again.

Your trading combo in lane is E+Q, if you can go into melee range you can also auto+W after that, this will proc electrocute or fleetfootwork that will make you run back towards your minions faster.
Your fastest full damage combo is: E>Q>R(stacked as much as possible)>W.
If you are chasing someone you usually have to start your combo with R then use E>W>Q/W>E>Q.
If you have Zhonya's you can R into enemy team and insta use Zhonya's to take enemy focus and make them waste some important abilities for a teamfight.

You are pretty weak before level 6. You should avoid fights at all during levels 1-5, but also you shouldn't let enemy laner push your lane and roam.
At level 6 you might start to look for some aggressive trades on lane, but be careful since your R is on 5 sec CD and that is a great oppurtunity for enemy jungler to gank you, don't use R forward if you don't know where enemy jungler is.
Your next Huge powerspike is when you hit level 11, your R is now on a decently low CD especially if you already have some CDR in your Items. At 11-15 you can already jump all around the map with no worries.
Next huge powerspike is once you complete stacking your archangel's staff. It grants you an enormmous dose of AP and an extra shield.
The moment you win the game is when you hit level 16 and complete Rabadon/Lich Bane. You just run and jump straight on the enemy team and one-shot every squishy target.

You need to abuse your runes:
-Make use of fleet footwork heal everytime you are able to do it without taking damage
-Spam Q (9sec CD) to continuously stack your manaflow(15sec CD)/heal from Taste of blood (20sec CD)
-If you max your E, it's a great tool for waveclearing
-Care more about farming minions than trading with enemy laner.
-Learn how to CS under turret since you will do that a lot while playing Kassadin
-Tell your jungler that you are pretty weak early game and that he should't rely on you.

After laning phase you should go sidelane (top/bot) in order to get as much EXP/GOLD as possible and farm as much as you can till you get your Lich Bane/Rabadon and reach your powerspikes. This works really well if your TP is up, however it's not neccessary when there's no risk of enemies taking an important objective on the other side of the map. Also you should look for farming jungle as much as possible at levels 13-16. You pretty much one shot every jungle camp at this moment. This also works very well if your jungler is an early game champion such as: Lee Sin, Xin Zhao, Hecarim, cuz they don't need that many resources late game.
When you reach your powerspikes you should look to end the game ASAP or the enemies will get stronger.
In teamfights you should look to flank enemy backline and one shot squishy targets. (remember about stacking R before that)