Since Ryze was revamped many people have been trying many different builds. This build is what I use whenever I play Ryze. I do very well with him because he matches my playing style. This means that I am asking you not to down vote this because it does not match you. Ryze has changed from a classic mage to a champion that has a couple sides to exploit in the same build. That build is this one.
This will be a wall of text and I apologize. It will be updated with pics and so on to make it more than just text and a great guide :).

Pros:
Great damage burst
Good attack damage
Great team asset
Can do more than just burst damage
Rune prison is like a stun
Spells are based on AP AND MP

Cons:
Fairly squishy
Targeted in team fights
Relied on for rune prison

I am not a very big person on runes. I have a couple that help with other characters that I play but I do not have any for this Ryze build and I do just fine. This part is completely up to you. I personally believe that runes are only helpful in the beginning of the match so I focus more on play style and my experience with each champion.

This is the part of the game that makes me want to pull my hair out. I love putting 21 into one of the three areas but sometimes it just doesn't feel right. I normally play with 9/0/21 but every once in a while (depending on my fellow champions) my masteries look like 21/0/9 (this is better understood at the end of the build).

Other than your skill with this game and its champions the most important part of this game is your item selection. Every person has their own way of buying items and putting them to each champion.
Now is where people will have some problems with this build. I again ask that you do not down vote if it does not match your style or before you read the whole guide and try it for yourself.

Core Build:
Sorcerer's Shoes
Manamune- reason why is that mana is great and this gives a lot but also it gives you a good attack damage. Some people have said this is stupid but my auto attack has gotten me so many kills that this is a great item especially early game.
Archangels Staff
Rabadon's Deathcap
Lich Bane
Frozen Heart

Now as any person that has played more than one game of LoL will know that not every game goes as planned. You have to change up your build a little so for those pesky games here are some situational items.

Situational Items:

Rod of Ages- I used to love this item but it just didn't do enough for me after i played for a while. But in some cases I need the extra health to stay in the fight longer or just not die as fast. Of course this item would replace your Frozen Heart but would be placed before your Archangels Staff.

Rylai's Crystal Scepter- for the possibility of needing more health.

A quick thing on why the skills are in the order they are in. First off your priority should be DP (desperate power), Overload, RP(Rune Prison) and SF (Spell Flux). The only exception is that at level 5 you take SF so that you have all of your abilities for the maximization of your ultimate's spell vamp.
Once you hit lvl 7 changed your priority to rune prison and then overload. RP is very key in any fight with a champion(s) thats why once you hit lvl 7 and change to mid game RP is more important.

This is where I am going to write nothing other than the reason why I am writing nothing. The reason is that every person has spells that they love and you know whether or not you can effectively use some spells. Therefore this part is all on you.

This is the part that you can either own or help your laning own. I normally choose to let someone else take mid so that I have a partner to help me own our lane. the second you enter the game buy a sapphire crystal and two health potions and head to your lane. What you want to do is harass the person that doesn't have a health regen item. Thats pretty basic. You just watch your mana and focus them as much as possible. Once you hit level 3 let them push a little and take down whoever has the lowest health, First use Overload then RP and then overload again auto attacking when necessary. that should get you at least an assist. Keep this up until you hit level 6 and then you move on to mid game.

Items order:
Tear of the Goddess
Sorcerer's Shoes
Catalyst the Protector
Glacial Shroud

Now that you have your ultimate you can begin running around if you want to. This is not necessary but if you choose not to you have to take towers (thats what wins games). When ganking you want start with RP, then SF, Overload, AA, then just whatever else you have at the ready. Use RP as much as possible to keep them from running or maybe chasing if something went wrong. Basically this is what you do until the big team fights start then you move to late game.

Items Order:
Manamune
Tear of the Goddess (archangels staff if you have the money)
Rabadon's

Now your team needs you to fight 5 on 5. This is where you need to be careful. If you have done as well as i normally do then you will be focused. That means stay in the back of the fight and focus the squishies so they don't kill you or your team mates. Once the fight has started it ends almost as quickly hopefully with your team on top. When the other team tries to run just RP and stop them. Once the other team is gone push, if you have the health for it.
You should have quite a few items by now but you still want to get up your gold and CS (creep score). So go push lanes when you can and kill anyone that decides to venture off alone. Only do this if your certain you will win and that it is not a trap.

Items Order:
Lich Bane
Frozen Heart

I hope this build works for you the way it does for me. Try it before voting. Screen shots of games will be coming soon and if I left something out just tell me and i will update as much as possible.