June 21th, 2011:

I've heard that Expanded Mind doesn't work correctly, it only increases the max of your base mana. There are conflicting claims by Riot that the skill is broken and that the tooltip is broken. So, until they fix/rework it, spend your points elsewhere.

What's worse than a fed carry? A fed carry that can't be taken down. Ryze has access to very powerful items due to the unique working of his skillset: the more mana he has, the more damage he dishes out. And mana is a pretty useless stat for most champions, so the items that contain this stat are always coupled to other, very powerful, stats.
This guide will show you how to get the most out of Ryze using that knowledge.

Unlike other Ryze guides, I don't focus on some minor extra damage by pumping up his mana to extreme numbers and getting massive mPen on him (or, for some reason after his rework, still focus on AP), but I take Ryze's heavy damage and make him unbreakable.

Greater Mark of Magic Penetration runes are universally good for casters, Ryze is no exception there. So, we have the marks covered.

With the given build (items, masteries, skills and runes), you get 36% cooldown reduction, get 9 Greater Glyph of Knowledge runes or 9 Greater Glyph of Mana runes if you are sure you can get blue buff or elixirs. If you're not secure about getting the blue buff or elixirs for some reason, switch out 4 for Greater Glyph of Scaling Cooldown Reduction runes, which puts you at 39,6% cd. The last 0.4% cd is negligible.

9 Greater Seal of Knowledge runes or 9 Greater Seal of Mana runes - just get 9 of these. They might be secondary runes, but they still give decent value.

3 Greater Quintessence of Knowledge runes or 3 Greater Quintessence of Mana. Ryze scales with mana, remember?

Scaling or Flat mana runes?

As you've seen, I offered a choice on which mana runes you get, I prefer to go with scaling mana runes as Ryze is a carry anyway and needs to play it safe till he gets beefed. But for a more powerful earlygame, especially for level 1 teamfights, you could swap out the scaling mana runes for flat mana runes. Flat mana runes will give you 290 mana (including Expanded Mind), which means you'll end up with 386 mana less at level 18. The turnaround is at level 8, where you get more mana from scaling mana runes.
Either way, go either for scaling runes or flat runes, not both.

7 points in Defense - because you really want the Strength of Spirit bonus on Ryze, he's a walking mana orb.
21 in Utility - maxing Expanded Mind is important, for the rest it's logical. Quickness , Intelligence , Utility Mastery and Presence of the Master are all very good on Ryze.

I choose to put my remaining 2 points in Archmage's Savvy . Feel free to place those anywhere else, not a big deal, they're spare points. I will clarify why I won't focus on Archaic Knowledge (15% mpen) - you already have Greater Mark of Magic Penetration runes and Spell Flux to decimate magic resistance and %mPen reduction gets applied after flat mPen reduction. So you'll be reducing 15% of nearly nothing, there are far better choices to focus on.

Now for the fun stuff, the item build. The exact sequence in which you get them may vary, depending on the line-up of your enemies.

Core items

Ryze needs these items, he's useless without them.
Tear of the Goddess
Get Sapphire Crystal first, it gives you 2 health regen from Strength of Spirit and your Overload hits harder with it. Return early to finish the item, as you need to build up charges asap.
Mercury's Treads
Wait, what, no Sorcerer's Shoes? Indeed. There's no point in being able to dish out high damage if you're silenced or stunned for the duration of the fight. Tenacity is huge. Besides, Spell Flux and Greater Mark of Magic Penetration runes will make short work of their magic armor. If you feel you don't need Tenacity, consider getting Sorcerer's Shoes, Mobility Boots or Boots of Swiftness.
Glacial Shroud
This is a no brainer, it gives you 15% cooldown reduction and a huge boost to your mana. Finish it into a Frozen Heart, if you're struggling against physical damage heroes. Otherwise, leave it until lategame.

Secondary core

Feel free to build one luxury item before finishing secondary core.

Will of the Ancients
"But it gives no extra damage!" Indeed it doesn't, but by the time you finish this you should be sitting around 4k mana, you're hitting harder than Duke Nukem on steroids at a boobslap convention. Why not take advantage of that and greatly add towards your survivability? It also sustains your team, meaning you're an even bigger threat.
If a supportive teammate gets this item, thank him, build the Hextech Revolver and laugh your balls off during the next teamfight.

Luxury items

Well, I tried to keep the core as short as possible, so you have a lot of choice to deal with varying circumstances. You don't need to finish all your core items before choosing one of these, but you should really aim to have at least your primary core items.

Rod of Ages
It's not core per se, but it's highly recommended getting at least one of these early on. You use everything on this item, even the AP. They stack, so you might consider getting two. Make one early, before your Will of the Ancients and maybe even before Glacial Shroud, depending on the situation.

Archangel's Staff
But Ryze scales poorly with AP! Yes, yes he does. But you already have a fully charged Tear and you should only build this when you're around 4.5k-5k mana, so why not take advantage of it and for only 1860 gold and no item slot get around 165-185 AP? Let's not forget the 50 extra mana either.

Banshee's Veil
Why is this not in Ryze's core? Well, the enemy will just knock the shield off with a Gust, Bouncing Blades, or some other low-damage spammy spell and it will be useless for another 45 seconds. It's vastly inferior to Rod of Ages. All stats on it synergize with Ryze though and it works well against a team with lots of single target spells.

Frozen Heart
This gives you extra survivability versus heavy auto attack champions, a little extra mana and it costs you no slot since you already have Glacial Shroud. Finish this earlier if you're up against an AD team.

Lich Bane
It gives you an extra nuke and with the very decent mana boost and the extra movement speed, it's a great addition to your arsenal. Don't try to get all procs, just auto attack when all your spells are on cooldown.

Rylai's Crystal Scepter
Gives a nice health boost and slows down enemies. Pretty nice, especially since you can really spam your single target nuke. Too bad it doesn't do much to increase your damage output.

Hextech Gunblade
Stacking Spellvamp works nicely on your sustained heavy magic damage output. The active also works very nicely for Ryze.

Thornmail
You do have a lot of health and armor, so why not? Only take this against extremely hard hitting auto attack champions though (at least 3 physical DPS champions).

Zhonya's Hourglass
Only take this against heavy AD teams. If you're being focussed, after having fired off all your nukes, activate it. The armor is useful too ofcourse.

Quicksilver Sash
It works well to counter specific line-ups.

Elixir of Brilliance
Let someone else get the blue buff, you'd only need it for 4% extra cooldown, which you can also get from this potion.

Fail items

Don't get these on Ryze, they're common pitfalls.
Rabadon's Deathcap
You really don't need AP on Ryze. Build mana on him if you want to do damage - mana items are also coupled to decent bonuses, unlike this item, which just gives a ton of AP and nothing else.

Void Staff
For the same reason you didn't get Archaic Knowledge . You have Greater Mark of Magic Penetration runes and Spell Flux to reduce magic resistance by a flat amount. %mPen applies after flat mPen, so, you'll just be reducing a nearly non-existing magic resistance. Sure, it's not bad as you'll do extra damage, but there are so many better choices. Just ignore this one.

Deathfire Grasp
You don't need any of its stats anymore and you don't build AP, so it's active is not that great. Besides, with your damage output you'll even lay waste to enemy tanks.

Get Spell Flux first, unless you focus on early teamfights, than get Overload first. "What, Spell Flux first?" Yes, even a single bounce gives you 50+50 damage for 60 mana, while Overload does much less damage for more mana. Ofcourse, if you need to dish out everything you've got at level 1, take Overload first. The low cd on it gives a higher overall DPS.

Max Overload first (its mana cost doesn't increase), 1 point in Spell Flux (high damage for a single pointer if it bounces, plus it lowers magic resistance and is good for a combo), get 1 point in Rune Prison at level 3 (for snaring and damage) and max it after Overload put 1 point in Desperate Power at level 6. Desperate Power gives high damage and its extremely good for farming and giving you a health regen boost, but it scales horribly. Wait with maxing it till level 18.

I went for Ghost and Flash as summoner spells, but there are definitely other viable options.

The good

Flash - Great for escaping, great for flashing in and focussing the squishy carry, it's an allround useful and overpowered tool. Try to waste Ghost before Flash, as Flash is much more versatile and useful.

Ghost - useful for escaping or getting your snare off - a snare can easily turn into a kill. You need a movement advantage though, as any player with half a brain will stay away from your snare range.

Exhaust - works very well with your skillset. An extra snare, so to say. You still have points to get the upgrade as well.

Ignite - you definitely need some in your team, but I think it's better to leave this to the non-carry nukers. You don't have access to its mastery skill either.

Clairvoyance - awesome skill, but it's supportive, try and get a supporter on your team to take this.

The bad

Teleport - useful if you're in a solo lane and good to get in a fight if you're farming on the other side of the map. I personally prefer dual lanes though.

Smite - you're not a jungler

Cleanse - not bad, you will be focussed heavily after all. It's too defensive for my taste though.

Heal - you get enough instaheal from your spellvamp.

Fortify - leave this to the tank.

The ugly

Clarity - you'd think getting a big mana boost on Ryze would work. However, Ryze's mana thirst is insatiable, the amount of mana you get out of it is nearly negligible.

Revive - not a big fan of this skill, I don't like speccing to failure.

Rally - this one deservers the ****piest summoner spell of the year award.

Ryze can go into a 2v2 lane or a 1v1 lane, in a 1v2 he can be effectively shut down. I prefer to take the 2v2 lane so someone else can profit from mid and Ryze can take the blue buff without endangering his lane. Go back early to finish your Tear of the Goddess and start building charges. Once you have Tear, start farming minions with Overload. Play it safe until you get beefy. You are very squishy earlygame. You are the carry, you owe it to your team to play safe.
Don't be afraid to spam Desperate Power if you're farming. The cooldown is low and because of your passive it will recharge even more quickly. You can also use Desperate Power to push out a lane to go ganking/scouting.

Don't be afraid to use Ghost to get your snare off. Only waste Flash if you're sure you can get a kill with it you wouldn't have gotten otherwise.

From level 6 and on, try getting dibs on the blue buff. You have a pretty big claim to it, as it'll help you tremendously.

Remember that you're not a dedicated tank, but you can still soak up a lot of damage.

The pros and cons of this specific build:

Pros

- Carry, tank and weak initiator all in one
- Benefits his team by just being around

Cons

- Weak early game
- Needs lots of farm
- Silence can shut down your damage output effectively.