Let's look at his skills first.
Arcane Mastery (passive)
Every time you use a spell the cooldown of all your other spells get's reduced by 1 second. Pretty good passive, not much else to say.
Overload (Q)
Single target nuke on a decent cooldown. Also gives 2% Cooldown reduction per level. 0.45 AP ratio. Scales with 10% of your maximum Mana.
Rune Prison (W)
Single Target root. Not a stun, this is important. People can still kill you while in this. Also deals damage over time while the target is rooted. Relatively long cooldown and Ryze's only CC spell.
Spell Flux (E)
Bouncing type spell. Bounces up to six times. Every bounce deals the same amount of damage and reduces the target's magic resistance. Decent spell for farming and harassing
Desperate Power (R)
Gives Ryze decent bonus AP and all his spells do 65% AoE damage for 8 seconds. Quite a short cooldown for an ultimate. With this you can kill a whole team if they clump up.
Skill order:
Get one level of Q E and W first. I suggest starting with W then getting E and Q last. After that max W first because it does quite good damage and the root time also increases per level. I suggest maxing Q after that because the damage difference of leveling Q and E isn't that big and Q also gives you 2% CDR per level. However some people also like to max E before Q. You should get R every time you can.
Kill Combo:
R > W > E > Q
Summoner Spells:
Ghost/Flash
Masteries:
9/0/21 or 0/9/21. Some people like the damage bonus with Archaic Knowledge, others like the very powerful regen with Strength of Spirit since Ryze can get quite a lot of mana. I'd say you test both for you.
Runes:
Standard caster setup i'd say.
Seals: mp5/level
Glyphs: CDR
Quints: Health probably
Marks: Magic penetration
Item build
You can do different builds for Ryze. Some people like it very mana and HP heavy, other's like full out damage. I like a balance between those two.
Item suggestions:
Boots of Sorcery/Mercury threads:
Depends on how much you need survivability. Against a halfly competent team with some CC I suggest Mercury threads most of the time, but it's your thing, really
Rod of Ages
I really suggest this on Ryze as a core item. It gives you some HP so you don't die as instantly, some nice AP and some mana you need and it also gives your Q more damage.
Mejaj's Soulstealer
You should really get this as fast as possible most of the time. Unless you or you team really doesn't do well. If you almost never do well enough for Mejaj's you probably shouldn't pick Ryze in the first place.
Archangel's Staff
A lot of people like to rush this as it gives you good AP, enough mana for the rest of the game and TONS of damage for your Q. However you'll die like instantly without any HP item so I rather suggest getting a Tear of the Godess early and completing the staff after you got some HP.
Zhonyas Ring
As for most casters a very good item for later in the game. Also the 2 seconds from the active are better than a bit of HP because you die so fast anyways. Without doubt a very good item on Ryze, even if he doesn't have the best AP scaling on the planet.
Void Staff
Very good damage item for later in the game, since people usually start stacking up some Magic Resistance and Void Staff helps do deal with this and also gives good AP.
Rylai's Crystal Scepter
The slow isn't too strong on Ryze in my opinion, but a good chunk of extra HP and good AP are always welcome. A good item for very late in the game.
Guardian Angel
Very good lategame survivability item, just like for most heroes.
Playstyle
On lane, farm and harass with W and E early on. If you have everything ready and with help of your mate you can proably kill something. In mid and lategame positioning is REALLY REALLY important on Ryze. It makes the different between you being alive and their whole team being dead or you being dead and they being alive. Use R when they clump up, get your WEQ combo out and run until your cooldowns are ready again, then nuke again.
If you have any suggestions, criticism, ideas on my guide or ideas for discusison on Ryze, please post.