I have play with ryze since I was level 1 (now lvl 29.5).
In the beginning I didn't know the best way to play this squishy guy, but now it's obvious to me. First of all (spells) you need to unlock Rune Prison and Spell flux before you have your ulti. Next up is to gain AP very fast.
As you can see: you will have nearly 700 AP at lvl 18, then you will pwn. Still squishy, so don't haste your kills. Time your ulti right. First Overload your enemy, then pop your ulti, get in range (close range) snare them with rune prison and get spellflux goin on. Then you need to run! Unless your team is with you and have enough DMG.

It's important to start with Ability Power. When you start with movement speed you don't have enough damage to get kills, and when you are stunned, you die.
Get 2 Amplifiyng thomes. Get shoes and then sell the second thome and buy the Blasting wand. When you have 1600, immediantly buy the Needlessly. Then you have 1000 left for the Rabadon's deathcap (makes you getting something like 200/300 AP. With your ulti it soon will be 400.
Then get your sorcerer. When your build is finished you will have a thome, sorcerer's, 2 rabadons and 1 needlessly (400 + 60%). Makes you have 696.1 AP and with your ulti (+120) you have 820. When you stack your kills you can even buy the Mejai's for 20 stacks+180 AP.
Then you are done. No one can kill you unless they are with 3.

And I have tried this. The final minutes (53:00) I had base AP 790 and with the ulti i had nearly 920. Spellflux when 3 enemies are near and you have a sweet tasting triple.

Farming:
Farming in early game is good. Use spellflux for getting a lot of minions and in that way stack some gold. In that way you can easily gain more AP without any kills. Don't overdrive and don't rush into enemies when they have low HP. They will always try to gank you as you are already squishy.
Getting the last attack on minions with spell flux gives you lot of XP and gold. Make sure you are always backed-up by a ally.

Good luck *check my other build for miss fortune!*