My first guide, so any feedback on how to make it better would be awesome!
This guide is meant to be useful for newbie players as well as the vets, so there might be some things in it that more seasoned players would consider common sense, but I'd rather have them there for someone to skim than not have something there because I assumed someone knew it.

I really feel like Kassadin is one of those champions who is really underrated. He's got more mobility than just about any champ out there, a silence, a slow, and with his recent change to his W skill he can melee for a good chunk of damage, even without any AD. Not to mention he's got to be one of the coolest looking champions in the game, why wouldn't you like him?

Pros

Best mobility of any champion
Has a silence with good range and damage
Able to tower dive and retreat easily
A good aoe slow to help teammates catch up to runners
The only thing that can kill you is crowd control which prevents your escape
Looks amazing

Singed's tears are delicious


Cons

Not a whole lot of damage beginning game
Low base hp
You WILL be focused

I run the 9/0/21 build you can see at the top. The cooldown reduction and bonuses to mana and regeneration are just amazing for Kassadin, especially with a mana intensive build like this, and the extra exp is awesome. The extra movespeed isn't essential for Kassadin, since his mobility comes from Riftwalk.

For runes, I build Greater Mark of Magic Penetration, which are always helpful for a mage, Greater Seal of Scaling Mana Regeneration, because you only really start using your mana once you get your ult so I found the per level stuff to be much better, and Greater Glyph of Ability Power with Greater Quintessence of Ability Power, to help out Kassadin's relatively weak damage output in the early laning phase.

The quintessences aren't a must. I've played with a couple different types, and as usual, those are mostly personal preference. I just find the AP ones to work best for me.

Recommended Summoner Spells:


Ignite - Great for Kassadin, simply because after his big burst following his teleport you might find yourself a tiny bit short of that kill, and ignite can net it for you quite often.


Flash - A lot of people don't like Flash on Kassadin, saying that he's already got a built in one, why use up a summoner spell? Well I love it because it's great for when your teleport is falling just short of your fleeing target, or when you have no mana to Riftwalk, or when you just REALLY need to get away from that Yi with the 800 movespeed and your teleport just isn't fast enough.

Other useful summoner spells:


Cleanse - It's useful for escaping cc heavy teams, seeing as keeping Kassadin from using his teleport is the only reliable way of killing him short of a one shot, but my banshee's veil usually allows me to escape the initiating cc, and of course, if your enemy has a cc heavy team, you should be getting the Mercury's Treads which will also be really helpful with escapes. Cleanse would be my third choice, after Ignite and Flash.


Clarity - I really don't think you need this if you have the mana regeneration runes, but if you're new to the game, and don't have the runes yet, then this might be helpful.


Clairvoyance - Should really be taken by a support instead, but it's still a useful spell, especially in ranked. Just remember to configure your masteries for the improvement if you end up getting it.


Teleport
Some people like to take teleport to get back to their lanes faster, but with the speed that Kassadin can move around the map, I prefer to take spells that help during combat itself.

Summoner Spells To Avoid:

I don't really like Ghost, because movespeed won't help Kassadin nearly as much as something like another teleport with Flash will. Usually if you need to use Heal as Kassadin you're probably going to die anyways. Kassadin already has a good slow, and you probably won't ever be within range of most melee champs anyways, making Exhaust low on the pick list.

For your harassment, you should simply shoot your Null Sphere at them whenever they come within range, then moving back so the sphere will collide with them before they can enter range. This prevents them from hitting you with any spells as well as doing a good deal of damage.
Once you get your Riftwalk you can do this much easier. You can teleport on top of them, instantly hitting them with Null Sphere when you land, and then slam them with Force Pulse. This does immense damage and leaves them silenced and slowed. Essentially leaving them crying on the floor whilst clutching their groin while you walk back to your side of the minions, since they cannot cast, and they cannot chase.




Null Sphere (Q)
(Active): Kassadin fires an ethereal bolt of void energy, dealing 80 / 130 / 180 / 230 / 280 (+0.7 per ability power) magic damage and silencing the target for 1 / 1.4 / 1.8 / 2.2 / 2.6 second(s).
Cooldown: 9 seconds
Range: 700

Kassadin's silence is what we'll be maxing out first. It has a great range, good damage, and the animation to start casting is very quick, so you can move into the max range, and the instant he starts firing, you can move back away from your enemy, so they can't harass back. It also has a pretty short cooldown, so you can use it to last hit minions, if you really can't afford to get into melee range with them. Always remember to cast this before anything else after you teleport next to an enemy. This keeps them from being able to fire anything back in response (like a stun) while you unleash your combo.




Nether Blade(W)
(Passive): Kassadin's melee attacks restore 8 / 11 / 14 / 17 / 20 mana. This effect returns triple the mana against champions.
(Active): Kassadin's normal attacks deal 20 / 30 / 40 / 50 / 60 (+0.15 per ability power) bonus magic damage for 5 seconds.
Cost: 25 mana
Cooldown: 12 seconds

This move recently got a buff so that it's more than just a way to charge Force Pulse. It now adds magic damage to your attacks, which gives you a reason to chase and hit people with your fancy purple lightsaber rather than flailing around behind them until your cooldowns reset.

This ability is also useful for adding that last little bit of damage to your attacks if you're having poor luck with last hitting minions. I strongly encourage you to practice last hitting, as it's an important skill to have so you don't push your lane too far forward, especially with a squishy like Kassadin.

The mana regeneration passive on this ability is also nice, but not nice enough for you to auto attack minions for mana (unless you are bone dry, and there is really no choice).
Those effects are nice, but my favorite part of this ability actually isn't in the text. My favorite part is the low cooldown and mana cost. This means you can use it quite literally every single time it's up, and this helps in a few ways. First, it gives you a charge on your Force Pulse, which is awesome, because it adds a lot of damage to his harassment, but you don't want to be caught in a gank without your slow, so by using Nether Blade, you keep it up and ready. Its second effect is that it also counts for your Tear of the Goddess. Every time you use this extremely low cost ability, you gain 4 mana to your maximum total! 4 might not sound like much, but it adds up to give you 1000 mana maximum after a while if you use it every time it's up (or at least most of the time), and this adds to the AP Archangel's Staff gets you later.

We put one point into this at level four, which should be right about when you get your Tear of the Goddess, and then use it for the charges and mana with that one rank, only leveling it up when it's the only skill left.




Force Pulse(E)

(Active): Kassadin draws energy from spells cast in his vicinity, gaining a charge whenever a spell is cast near him, including his own spells. Upon reaching 6 charges, Kassadin can use Force Pulse to deal 60 / 115 / 170 / 225 / 280 (+0.7 per ability power) magic damage and slow enemies in a cone in front of him by 30 / 35 / 40 / 45 / 50 % for 3 seconds.
Cost: 80 mana
Cooldown: 6 seconds
Range: 400
Cone Width: 90º

A wide arc of pain and slow, this ability is great for chasing or escaping, and it has a very short cooldown, but it needs 6 spells to be cast near Kassadin before it can be cast. This means that the spell counter restrictions will most likely be what keeps you from using this often in the early lane phase, but in teamfights, it is up VERY frequently. This spell can be cast through walls to slow enemies chasing allies, or shot toward a fleeing enemy to slow them enough for your allies to reach them, and collect that sweet, sweet money.

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IMPORTANT

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One thing a lot of people don't do with Kassadin is use that force pulse counter as a detection ability. Have you ever been just standing alone in your lane, and all the sudden the counter jumps from 3 to 4? Somewhere around you, someone used an ability. If none of your allies are there, then that means an enemy is probably in that bush near you, preparing their buffs to strike, or in the jungle right above you, using abilities to farm the minions there. This is why I like to try and keep the counters somewhere between 3 and 5 if I can. This leaves enough counters on so that you can get it ready instantly with your combo, while leaving it low enough so you can see abilities used around you.

Kassadin's slow has the same ability power ratio as his Null Sphere, and casting both after Riftwalking into melee range of an enemy champion is sure to cause a good deal of pain. We get this to maximum level after Null Sphere, making our opening combo do its maximum damage by level 13.




Rift Walk(R)

(Active): Kassadin teleports to a nearby location, dealing 60 / 90 / 120 (+0.5 per ability power) magic damage to surrounding enemy units. Each subsequent Riftwalk in the next 7 seconds costs 100 additional mana and deals 60 / 90 / 120 additional damage. The cost and damage can stack up to 10 times.
Cost: 100 mana (+100 for each teleport after the first within 7 seconds)
Range: 400
Radius of Damage AoE: 150
Cooldown: 6 / 5 / 4 seconds

Kassadin's signature ability, his teleport. This ability allows you to move around the map at incredible speed, making him the best champion in LoL at chasing and running away. With the added burst forward from Flash, you can clear huge distances in an instant, or going through two walls without having to wait for the (extremely short) cooldown on Riftwalk to come back up. At maximum level, the cooldown is about somewhere between 2.4-3 seconds, depending on which items you got throughout the game. This ability also makes tower diving a breeze. You can dive in, burst most squishy champions from about half hp, and then teleport out, maybe taking two shots from the turret before you're out of range.

Obviously, we take this skill every time it is available.

Meki Pendant and two Health Potions
This gives you some mana regeneration and emergency health to stay in lane longer, letting you harass freely with your Null Sphere, and get to your Tear of the Goddess faster.



Tear of the goddess
This item is VERY important. Get it as soon as you can, and once you have it, always remember to spam your Nether Blade skill (W) every time it's up unless you find yourself harassing so much that mana is becoming a problem. It's a low cost, low cooldown ability that makes you hit harder, which also makes it easier to last hit if you have trouble with that. This should build up to maximum stacks really quickly if you're diligent with your W and use your other abilities liberally. It also keeps it's stacks when it eventually upgrades into your Archangel's Staff, so it's not a big deal if it's not upgraded right away.



Sorcerer's Shoes or Mercury's Treads
Next you want to get boots. Boots aren't a static item that you buy every time. You should buy Sorcerer's Shoes if you feel like you need a little extra damage, or they start stacking magic resist, or Mercury's Treads if their team is really heavy cc. If the enemy team has 3 or more ways to prevent your teleport, with things such as stuns and silence, you should definitely get the tenacity.



Morello's Evil Tome
This is an amazing item for Kassadin, and I'm surprised I don't see more people with it. It has cooldown reduction, ability power, and mana regeneration, and it's not too expensive. Everything you want is in this nice little book.
Edit: After some playtesting between these two items, I believe you should only get the tome until you feel you are good enough with Kassadin to chain the active ability of Deathfire Grasp into your combo. After you become proficient at that, your damage increases immensely.

OR


Deathfire Grasp
I don't really like using this item, but that's mostly because I prefer not to have items that aren't always doing something for me. If you like this item, then I would get it instead of Morello's Evil Tome.
I'll mess around with this later, and see how I do with this item compared to the Tome.
EDIT: Tried it out in a couple games and compared how I did with my games where I used the tome, and it was about even, although this could just be me getting used to having it, and being new to using the active. I still think it's up to personal preference, but I think I'm going to start using the Deathfire Grasp more than Morello's Evil Tome now, as I can see that becoming the better pick once you become exceptional with chaining its active into your combo. Just always remember to cast it first, as it does a percentage of their remaining health, so if you hit them before using Deathfire Grasp, it does less damage.
Another update: After playing more with Deathfire Grasp, and getting used to comboing the active into my chain, I believe it is better than Morello's Evil Tome in just about every situation. The only way I recommend Morello's Evil Tome now is if you don't feel comfortable with an activated item.

CAUTION, ABILITY TO THINK FOR ONESELF REQUIRED AHEAD.

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(Acceptable to finish Archangel's Staff first, if you feel you have enough survivability at this point, but I recommend one of these first in most cases.)

This section is where you look at the enemy team, and what characters they're playing, as well as how well each one of them is doing, will affect your next item decision. Banshee's Veil is going to be the choice for most of your matches, since you will almost certainly be the focus of everyone's cc in teamfights. The other two are really only recommended if the enemy team has VERY little in the way of crowd control.

LOTS OF ABILITY POWER AND/OR CROWD CONTROL


Banshee's Veil
Gives you mana, health, magic resist, and best of all, that awesome spellshield, which will make it a lot harder to surprise you and pin you down with cc. This item boosts your survivability a ton. This is going to be what you get most of the time, because cc is Kassadin's biggest weakness.

NOT MUCH CC, AND AN EVEN SPREAD ON ABILITY POWER AND ATTACK DAMAGE


Rod of Ages
A good replacement for Banshee's Veil if the other team is completely devoid of almost any crowd control. It gives a lot of hp and mana, and it gives a nice chunk of AP too.

NOT MUCH CC AND/OR LOTS OF ATTACK DAMAGE


Frozen Heart
If the enemy team has a ton of AD, and they're just tearing you up, this is a good item to get. Cooldown reduction, mana, armor, and it slows their attacks to boot! This item is another good replacement for Banshee's Veil if the enemy team doesn't have a lot of powerful AP.

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Archangel's Staff
By the time you finish your Banshee's Veil you should have your Tear of the Goddess just about maxed out, and converting it into an archangel's will give you a massive boost to your damage and mana regeneration.


Rabadon's Deathcap
This will put your AP over the edge, so you can just annihilate your enemies before they know you're there. Some people would go after this much earlier, but I think that Kassadin's weaknesses are how squishy he is and his high mana cost spells, so I take care of those first, ensuring that I can survive the fights while still dishing out awesome damage. One comboing the caster isn't as satisfying as the sheer chasing and escaping power the cdr and mana items give, not to mention your ult will stack damage as you use it, so mana does give you more damage, not counting the Archangel's Staff bonus.

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Depending on whether they build to defend against you or not.


Lich Bane
By this time, they should be running in absolute terror whenever they see a purple cloud of death appear next to them. This takes your huge amounts of AP and adds it to your attacks which already scale with your AP through your Nether Blade! I usually don't get a chance to work up to this item, but when I do, it's very satisfying.

Or


Void Staff
If they stack a lot of MR, seemingly out of spite for you, then this is a great item instead of Lich Bane. Basically throwing away one of the two of their MR items, if you wanna think about it that way.

Final Build:

1. Either or

2. or

3. One of the three:

4.

5.

6. or

Also:

sight wardsight wardsight wardsight wardsight wardsight wardsight wardsight wardsight wardsight ward

WARDS

You are a highly mobile champion who can clear minion waves with 2 spells. Buy wards.

In this section, I'll explain how you should be playing Kassadin throughout the game.

Early game:

This is Kassadin's weakest point. Before he gets his Riftwalk he is extremely easy to kill, so you have to be very careful about ganks. Be aware of who's missing at all times, know what summoner spells and abilities your opponents have so they can't get to close and destroy you. Stay near your minions, as that will discourage their attacks unless they want to take minion damage, and fire your null sphere at them whenever you get the chance. Always try to stay just outside of the range for shooting it at the enemy champion, because that means they can't shoot you either. Every time it's off cooldown, target them with it so Kassadin runs forward and casts it, then immediately retreat to your initial position before they can counter you. They will probably get an attack in before you do, but you'll probably be doing more damage, because auto attacks trigger minion attention, while Null Sphere does not.

Keep your force pulse up and ready, don't use it to harass. At the early stages of the game it wont do much damage anyways, so just keep it for the slow in case you need to escape.

Go back for your Tear of the Goddess as soon as you get enough money, unless you are in complete control of the lane, and you think you can force your opponent back to their base soon. In that case, save until you can afford Boots of Swiftness. After you get the Tear of the Goddess, be sure to spam that rank one netherblade. It'll translate into AP later.

Once you get your ultimate, you can really start your lane control. Just wait for them to come into range to shoot something at you, whether it's Ashe trying to volley at you, or Brand setting up a Pillar of Flame under your feet. As soon as that happens, Riftwalk right next to them and unload your Null Sphere and Force Pulse. This accomplishes a few things. First, it does a lot of damage. Second, it makes them waste skillshots, and by the same token, mana. Third, it puts this perpetual fear into them that you can just be right next to them, obliterating their health whenever you want, which results in them cowering far behind the minion wave, far out of range of exp and last hits. Be sure to do this every time they try to shoot at you, or if they get bold and start trying to last hit minions to make sure they know who's the boss of this lane.

Mid game:

A little bit after getting your ultimate, you should be king of middle lane, maybe with a kill or two under your belt, and you should have your choice of boots and a Tear of the Goddess, maybe either the Fiendish Codex or kage's lucky pick, depending on whether you prefer the Morello's Evil Tome or the Deathfire Grasp. This is when you should start roaming a little. Leave your lane to see if top or bottom lanes need a gank. Kassadin is great for this, since he can jump in out of nowhere, silence and slow. Leaving your enemies helpless so that your allies can beat them senseless.

This should gain you a few kills, or at least assists. Just don't tower dive unless you're absolutely positive you can kill them in one burst, and they can't interrupt you with something like a stun or a silence, followed up by a flash away from the turret before it can kill you.

Be liberal with your Riftwalk when moving between lanes. Don't get stacks on it, but feel free to use it once, then wait the 7 seconds before it wont cost additional mana. The speed at which you move when using that spell is phenomenal, and it makes calling mia on you that much harder. This is especially effective if you can get blue buff, which you should be able to solo fairly easily after about level 9 or 10.

End Game:

If you've been careful, and avoided death while killing enemies in an instant like a ninja, then you should be an all powerful killing machine by now. You should be able to jump in, take a squishy down to about half (or less) health in one burst, then calmly leave back to your team. Or ensure kills for yourself and your team with Kassadin's unmatched chasing ability, combined with his slow. If they have someone like Katarina or Malzahar on their team, save your Null Sphere to interrupt their ults. It's much more important than the damage you get off of it.

Just try to be everywhere at once. By the time everyone is level 18, you should have a ton of ability power and ridiculous mana, so you can rift walk whenever you need to get somewhere like a teamfight in a hurry.

Can't win em all! :P

I hope this guide helps anyone who wants to play the best Rift Ninja out there! Any feedback on things I can improve or opinions on how you would do something differently is greatly appreciated!