People think that Kassadin is a traditional assassin like Fizz or Akali where you try to dive onto the enemy AD carry and hope for the best, then claim that he has a bad lategame. He doesn't really work like that.
In a way, it's Kassadin's job to poke before a fight starts with his R+Q+E. It requires fast fingers to do this and you have to know when it's safe to do that-- otherwise, you may get initiated on. This combo can easily chunk an out of position squishy down to 50% even if the R doesn't connect, but that doesn't mean you can't do it on a non-squishy too.
When the fight engages, Kassadin should stay back and R+Q+E the person diving onto your backline and attack them with W. You have to autoattack as Kassadin. Seriously. He has so much sustained damage and if you're not using Netherblade, you will be crippled.
It's also possible for Kassadin to burst down an initiator (especially Amumu or Fiddles. You know how they hide over walls? R to them and start attacking then back off when their team comes) or disrupt their initiation combo with Q.
You also have to know how to use your Riftwalk effectively and know how much you can take. It's hard to explain this, but keep in mind that just because Kassadin is at low HP doesn't mean he's out of the fight. R+Q+E someone and then run the hell away at the end of a fight if people are low.
If an AD carry or AP carry is without the protection of their bruisers a few seconds after the team fight starts, you can switch to them (bonus points if you do it by going over a wall to assassinate them) but keep in mind that if you don't burst them down, you're in a very vulnerable position as you're squishy and Riftwalk is on cooldown.
TL;DR: R-Q-E, walk away. R-Q-E, walk away. Are you safe? R-Q-E-W autoattack spam. Being initiated on? Hit Q to disrupt the combo.
This is speaking as a Platinum IV Kassadin main. Not the highest but whatever. My 6 item core is usually Sorc Shoes/Rod of Ages/Rabadons Deathcap/Void Staff/Zhonyas or Abyssal or Warmogs/Deathfire Grasp. Pick up a Deathfire Grasp sooner if an AP or AD carry is ridiculously fed, right before or right after you get Rabadon's Deathcap.
In lane phase pre-6 be sure that you're not taking free autoattacks, retaliate with Qs if they're in range. Also seriously. Learn to last hit. I don't know why but Kassadins have ridiculously low cs even after level 6.
Also in midgame or lategame you can probably find some idiot overextending to farm. Just go kill him then do an objective. Kassadin is an amazing duelist late.