Normally speaking, on other champions I can go in-depth on certain champion match ups and explain what you can do vs champions to win them.
HOWEVER, with Kassadin I think I'm just going to try and explain your goals in lane, as he literally loses all lanes 1v1 unless the enemy misplays.
(This is all pre-6 and before you have blue buff, as the lane changes after that).
1. Ideally speaking, you want to let them push on you. Kassadin can farm under turret extremely well with his W so you don't have a hard time last hitting under turret. This also keeps you safe from ganks and you are pretty hard to dive since you have a silence, just be careful using it and then getting dove.
2. In the case your lane is pushed out and you can't do anything about it, you need to ask your jungler to help you reset the wave so that you may be able to farm with either Q or at least not get zoned.
3. Do your best to survive in lane, just don't die and wait for level 6/blue buff. Once you get those two things, your life became a LOT easier.
Okay, so far it seems like Kassadin has a hard time in every match up. Which is mostly true but this does not mean he cannot win lane in the case they misplay or you have opportunities to punish them for mistakes. Here's a few things you can try and do!
1. In the case you're vs Riven or something, forcing them to use their spells and harassing them with your E will prove to do a lot more damage than people expect.
2. At level 2, if you have Q W you're much stronger than people think. Don't be too afraid of melee champions if they start fighting you, you can win the fight depending on who you're versus. And if a range champion ever comes up in range, you better smack him silly.
3. In the case you have ignite, you can try and kill your lane opponent by spamming Q/E and then flash autoing with W. Otherwise if you have teleport, just look towards level 6 so you can start making plays.
With Kassadin, I don't play to try and win lane phase as it's almost impossible early game, I look towards the mid game and post level 6 era where you've become infinitely stronger than you are the first 5 levels. By all means you can try and play Kassadin to win the early lane phase but it's extremely hard and I recommend playing safe and waiting for your power spikes. Once you hit 6 you can spam your spells on them without worry of running out of mana because of blue and the dorans/catalyst you bought.