Laning Phase on Kassadin is probably his hardest stage of the game. Your Goal is to not die early and take every CS you can. Don't try to force any plays as time is in your favor, and not in the enemies'. Early Tear and Chapter can help you very much.
Into AD matchups you're struggling against i can recommend buying wardens mail first as it's only 1k Gold and grants you insane Survivability stats.

Jungle ganks are nice but not necessary. Most of the time your Jungler will have to 1v1 enemy laner before lvl 6 as you don't have the necessary abilities to help him.

This is currently one of the strongest phases for Kassadin. Level 11 is currently his strongest Powerspike and if you're even you can basically 1v1 anyone on the map. You're a good Sidelaner as you can get out of very bad situations even without summoner spells. Your ult is a great utility tool. It's also great for Engaging on Enemies.
In teamfights you want to try and play around the abilities of your Enemies. Try to bait out key Abilities like Crowd Control and ultimates With your Own ultimate, as you can buffer 90% of abilities with your Ult.

This is also a very good phase for Kassadin. At 6 items you can still 1v1 basically everyone. You have great survivability and can easily nuke Enemy carries while still getting out. Your tower damage is insane as you are an AP champ and wave clear is also good.
I would recommend not Spamming 640 mana ults on waves as it can get very expensive and you might die because of poor mana management.