Alright, now that you're here, lets get this out of the way. You're most likely thinking that this is really stupid, but just hear me out. Once you read the explanations, it will make sense. I put this build together, oddly enough, because I played a game and DIDN'T want to tank. So, I decided to try something that I had in mind for a long time, and it worked. So I bring you Tankadin, the new way to play Kassadin.

Pros:
Good survivability against all damage
Great skill set even without AP (silence, aoe slow, spammable teleport, armor pen)
Easy to get in and get out of team fights
Starts out like a normal Kassadin

Cons:
People may think you're trolling
Damage output is situational, but will most likely happen
Mana problems early

0/21/9 is an obvious choice for someone going tanky in most cases. I may reconsider some of the choices at a later time. But for now it works.

Armor Pen Marks: Obviously being that we're going to get most of our damage from melee, this is the choice to go.
Armor Seals: You can never really go wrong with armor. I split between per level and flat for a good balance of natural defense throughout the game.
Magic Resist Glyphs: Pretty much same reasons as for seals, but for MR.
Flat Health Quints: More HP at the start is a great thing, especially for a character who normally doesn't fill this role.

My choices (Subject for change)
Ghost: This is great for chasing and escaping. Yes, I know you can port ever couple of seconds, but sometimes for what you're doing it doesn't hurt to be faster to cover everything you need to do.
Exhaust: You're a strong disabler. Why not add another?

Other choices

Clairvoyance: Great for people who like to hide in bushes, or just for scouting the map. Always good to have at least 1 in your team.
Teleport: Good to stop back door attempts. Has many other uses as well, but I'd imagine you should know them by now.
Cleanse: CC's are annoying, this may save you more often than anything else.

Doran's Shield: Its a good starting item for the HP, defense, and HP regen that you need for the start. You can't go too wrong here.
Boots of speed: Cheap, and you need movement no matter who you are.
Tear of the goddess: This item is to solve your MP problem forever. That, and it will be used later.
Banshee veil: This is an obvious choice. The shield it provides is great for preventing some damage/CC. The MR is great along with your passive, this way you seriously reduce incoming magic damage. And remember, and magic damage you get hit with speeds up your attack speed.
Mercury Treads/Ninja Tabi: This is why the item section didn't have upgraded boots. This is typically team dependent. Obviously if they have a bunch of auto attackers ninja tabi may be better, but against high magic and CC based teams mercury treads are the better choice.
Manamune: This is your main damage item. Its only 1100 after tear of the goddress, making it easy quick to obtain. The damage you'll get will help you take down their carries better. This will be your job anyways.
Randuin's Omen: So this is your solution to auto attackers. Being that most of them don't do magic damage on hit, this is how you make sure they don't destroy you. Also, the HP and armor is absolutely awesome on this item. Definitely a great choice.
Soul Shroud: Seems like a weird choice, but its 520 HP with 10% CDR to your team. The MP regen isn't too big of a deal, but the other parts add to this build nicely. That and its cheap.

As for the last item, it's situational. Pretty much, Last whisper or black cleaver for tanky teams, maybe something with life steal, etc, but preferably something with some kind of damage. Atma's impaler would be nice if they are very AD heavy.

Early game, you're regular casting Kassadin with more survivability. Don't be afraid to actually melee other people since you do have the HP and some armor pen. As usual, you will have to chill until you hit 6 and can port. Build up enough money for tear and boots and head back once you do. It will help. Once you have these, its still basic Kassadin.

Mid game is when it changes a bit. You should have banshee veil by mid game, and working on a mana mune. Try to gank often, since you're Kassadin you'll always be good at it. Try to gank with other people until manamune is finished since by this point your skill damage may not be very scary anymore. The effects of them however is now what you're going for.

Late game you should have randuin's finished and maybe even soul shroud depending on how well you're doing. With this, auto attacks will be slowed when hitting you, and AP casters and most other tanks will speed your auto attacks. So you want to get hit. And with your gear, you can take some hits. Hopefully your team has an actual tank as well, otherwise you'll be first in. Otherwise, you're 2nd in or your 1st in because you're slowing them for your tank to catch up. Being this close in on the fight will pretty much make it so your E skill is almost always available, so use it as often as possible. Use your Q to stop those annoying channeling ultimates, or to just stop those AP champions from being useful for 3 seconds. Melee down the carries hard. The good thing about you is that you can get in and out very easily, and focusing the squishy characters is easy when they can't run. However, if you need to run, it is not hard for you either, as the same thing that put you in should be back by the time you need to get out.