Early on, I played Ryze the AP way, dominating people and such. Then when the remake came out, I was completely confused: How the heck am I supposed to do damage? His AP ratios were no longer any good, and so I switched champions. Then one game, I decided to play Ryze again. I got into the game, and looked at his spells again. Then I came to the conclusion that Ryze, no matter how weird, was a mana/ CDR caster. After trying many times, I found that Double Catalyst, no matter how strange is a VERY good choice on Ryze.

7/25/11-Made Archangel's non-priority. Thank you Elk Arkane.

Basic caster runes, almost.
MPen. You need this to crush MR.
Mana Regen. You need this so you don't run out of mana for spamming
CDR. Pretty normal, you can get Flat Mana/mana per level glyphs too, if you want.
Sort of need it to land you Rune Prison. Fortitude, Flat Mana, they all work.

I like the 9/0/21 because of the MPen from Offense and the great Mana regen from utility.
I take the Flash mastery, because that is what I take as one of my summoner spells.

I start with a Mana Crystal and 2 health pots. You can start with a blue pot too, if you'd like.

Some people ask me: Why not the Meki Pendant, oh good sir?
Mana Crystal adds damage to your Q. Does the Meki Pendant do that? No. You do not need to stay in lane for long on before your second buy. You want to rush that Tear for the stacks it gives.

First Trip Back:
You want to have 630 gold or 980 gold on your first back. You can spam your spells when you are near the gold mark to push and farm the lane before you go back.

From here on, you have to make good calls on what to buy and when.

These are the items you will get EVERY SINGLE GAME (unless they 20 minute surrender or you do.)
Tear of the Goddess
I love this item. When you get max stacks, you'll have 1.35k Mana. Easily the best item on Ryze, except for maybe Frozen Heart. Archangel's Staff upgrade makes it even better. The thing is, the extra mana for 1.85k gold ISN'T worth it. Thanks Elk Arkane.
Catalyst the Protector]Catalyst the Protector
You may scoff and ask, why do you need two? Because when they are upgraded, they give you GODLY stats(not really godly, but they are really good). 1 should be made into a Rod of Ages. Another mini-stack item. 725 mana and 80 AP at max stats. More damage for you and some extra HP to top it off!
The second one is built into a Banshee's Veil. 375 HP and Mana is not to be scoffed at. The MR just adds to this item's greatness. And then the passive just makes it a must get.
Very good item on it's own, 425 mana gives your Q even more oomph, and 15% CDR is always good. If you upgrade to a Frozen Heart, you get even more stuff going for you. 99 armor AND 500 mana. Don't forget 20% CDR.

So. You've gotten 5 GREAT items. What about the last one? So many choices, but what's best?
Rod of Ages Rod of Ages
What again? You must be crazy! But I'm not. It might not be the best choice because of the stack build-up, but it still is great.
Quicksilver Sash Quicksilver Sash
This item is good against certain team comps. Malzahar and Warwick on one team? You're going to need this badboy for sure.
Force of Nature Force of Nature
Good against 2+ AP casters.
Zhonya's Hourglass Zhonya's Hourglass
Solid item, use it's active when you are getting focused down. 50 Armor is great too.
Thornmail 3-5 AD's on their team? Get this guy, and maybe even Zhonya's too. 300 Armor is going to own them hard.

Well, I would be dearly sorry, but we can't let that happen.
ITEMS THAT SURROUND AUTO-ATTACKING
Have you ever noticed that your spells have a longer range than your AA? Well now you know. Do I need to spell out that you should just use your spells. Oh, but Lich Bane will be good for attacking after one rotation! Yeah well, better just kite them around, then kill them with another combo.
IF THEY COME CLOSE WTF DO I DO BRO?
Oh. Did they go through your 2 second snare and your full combo and not be below Half-hp? Well then, wait 1 second and Boom dynamite them again. Rinse and repeat.
ITEMS THAT INCLUDE LOTS AND LOTS OF AP
Oh, didn't I mention Ryze's ****ty AP ratio's before? And you can't put 2 and 2 together and get that ****ty AP ratio's make ****ty damage, you need to be in a mental institution, because sooner or later, that ****ty AP Ryze is going to become an even ****tier AP+Lich Bane Ryze.

Easy Stuff. Some people take W first though. I like Q because it's easier to last-hit. But it's all preference.

FlashGreat escape spell and chase spell
GhostTier 2 escape spell compared to Flash
TeleportGreat offensive spell, best used if you are going top
IgniteGood finish to your burst combo
Cleanse5 seconds in CC can't be good from anyone's health.

Ryze can do great amounts of burst, but also he has great sustained damage.

Burst Combo is as follows:
Q>R>E>Q>W>Q
If you seem to be playing against people you can't burst, you should try this:
Q>E>Q>W>Q>R>Q>E>Q>R>Q
This will give you a non-existent cooldown on Q and make you deal damage like a AD carry would.
You should be able to get the first combo quickly, like a Brand doing his full combo.
For this reason, you should use SMART CASTING. It works like Karthus' Q. You no longer have to do the "who do I hit part anymore". Just hover over the person. And since Ryze has no skillshots...

"Why oh good sir, do you use E before W?"
I do it to lower MR. It may seem like a pittance, but really it helps. A lot.

Ryze is now the spell machine gun. He can throw down spells as quickly as anyone, and he does the same damage as before, if not much more. The fact that he doesn't give a **** AP makes it more amazing. My guide to Ryze, the caster without AP ends here.

SageAdvice

P.S. Remember to give it a bump up or bump down, depending on what you think. Give me lots of feedback, so I can try things out!