Not Updated For Current Season
Going for lots of ap. I see a lot of builds that do not take advantage of being able to snare someone and it is very useful plus it deals damage. Lich bane is very useful if you simply ult snare reg attack then bouncing ball the enemy will most likely die. If they dont reg attack again lol.
I get the first part of boots so I do not move to slowly early game. I then grab tear of the goddess so I am not constantly running out of mana. Then I finish getting sorcerer's boots. Next I grab majis for building ap through ganking. Next I grab zonya's ring because of the massive ap boost and the 2 second immune mid game is very useful. Next I grab rod of ages because of its hp ap and man boost. I then get void staff so I can penetrate all of the end game players magic resist then finally for the combo of zonya's and archangle staffs ap boost to dominate all enemies.
It is a slow starting build usually in the beginning you need to play safe and just Harass by trying to get them to attack the turrets then snaring them. Once you get your nuke and bouncing ball you can be a bit more offensive. If they try and attack your tower kill the mobs around them by ult bouncing then snare them and watch the tower eat them. Use ignite if they are about to get away.
I played a game perfectly once only knowing one of my other team mates who was in vent. he played jax and of course I was ryze. At the very end of the game i hade 621 ability power. When I attacked it went up because of the staff. The max I saw it at was 1k+ that means that bouncing ball hits the target for 841.5 each bounce. That includes the aoe damage when using ult. The aoe damage was about 331.5. The snare was doing 2376 damage if they remain in the snare the whole time and are hit by the aoe damage which was about 936. The nuke hits for 1749 including aoe damage and that you used it last. Then if you so desire you can use ignite which is at this point just mean.
Harass and stick by turrets when they are attacking.
Snare them if they attack you with in range of your turret.
If situations get dangerous you can use flash to block them with a bush or wall.
Don't start pushing until you get your ult.
If they ever attack you and are near your tower ult snare and bouncing ball along should cause them to not only take lots of damage but get eaten by turrets. In certain situations like when you are going to gank someone you can snare walk next to and ult spell flux over load and kill them very easily. You still have very low health so be careful about charging in to gank if your 1v1. Also surprise ganking is not only great for this build but some what necessary. If you think you can kill a champion but they have an ability you hate go in once you have zonyas and just immune through it and kill them.
By this time you should have a fair amount of ap which you can use to gank or just push a lane entirely by yourself. In team fights make sure not to be at the front of the back or even to far behind because its to easy for ryze to just get dropped.
This build does not work if you have bad team mates.
Not dieing is key to this build once you get maji's especially.
It is crucial that you do what ever you can to have mana and be casting abilities all the time. Mana pots in the beginning and later you wont have to worry.
If you feel like your gonna get attacked or ganked make sure to fall back.
Watch out for silences and anything that will cause you to not cast for six seconds after ulting
Use spell flux on enemies close together especially if any way their bodies form a line. Lastly make sure to not get close to high dps characters cause they can one shot to three shot you very easily even with regular attacks.
last note. Having zonya's ring is useful because you can initiate the group fight if need be and you know your friends will help out. Just kill then immune while your friends attack, then use your flash to get away. Also ap goes way above what it lists in the stats here since the majis can add at max 160 ability power.
I have adjusted this build to accommodate what happens in higher level games with premade teams.
As always feel free to comment and rate just make sure to comment why you rated such way.