First of all, Why Ryze?

As a Mid, he can lockdown the enemy allowing for much more successful ganks from allies, permitting that you're not able to stomp the enemy on your own.
As a solo Top, While he lacks health regen for long term sustain, that can be compensated for easily with pots. As with mid, he can lockdown an enemy for easy successful ganks, or in the event that your enemy has no magic resist, stomp them himself.
As a Tank, Ryze can get very tanky while still building mass damage. Even moreso if your enemy team is solid AD. Stack up the Frozen hearts and see how squishy they get.
As a Dual Laner, He can either carry the lane, (and with any kind of healer tend to dominate a lane) or focus more on locking down the enemy as a support and letting his ally tear the enemy a new one.

He's functional in most team compositions, in or out of mid. Even without health regen, he can be a tough champ to kill, especially with good supporting teammates. If you find health regen to be an issue, a quick purchase of Warmogs Armor quickly remedies that.

Currently my favourite Ryze build that I've been playing with. I'm sure I'll tweak it with time, but for now, it's what I've enjoyed playing with.
It's a partially tanky but still heavily damaging build for Ryze.

Pros

Tanky: Ryze can build quite tanky without sacrificing much in the way of damage, permitting you pick up items with both defense or magic resist and mana. (examples are frozen heart for defence and banshee's veil for magic resist)
Crowd Control: Locking down an enemy champion can make for extremely successful ganks, or turn the tide of a fight very quickly.
Fast Damage Output: Ryze is definitely a burst mage who can dump a lot of damage in one fell swoop. Starting with his Overload, then dropping the rest of his spells, you can follow up with a second overload due to his passive reducing cooldowns every time he uses an ability.
Low Cooldown Time:Due to his passive, every time you use a spell, all other spells have cooldown reduced by one second. Considering his spells, especially overload, already have quite short cooldown times, his spells are often at the ready.
Cheap to build damage:Building up mass mana is a lot cheaper than building mass AP. Since Ryze's two most powerful abilities gain power based on his maximum mana, his damage output jumps quick.

Cons

Lack of Sustain: With no spells or abilities to give you health back, you rely on either potions, or health regen items for sustainability. This can be countered a bit through the Strength of Spirit Mastery, warmogs, or the spellvamp from his ult.
Item Reliant: Without items, or without a good build, Ryze deals minimal damage and is fairly easy to squish. A good build is DEFINITELY important.

I opted for all mana per level glyphs, marks, and quintessences, and flat armor seals to give him a bit of additional early game durability.
I've tried full mana per level runes, and found the extra 12 or so armor more helpful early game, than the 14 extra damage for overload or the 9.5 extra damage for rune prison end game.

Rush your tear of the goddess for obvious reason. Every time you cast a spell, you get more mana. Ryze deals damage based on his mana. Enough said.

I grab basic boots and, depending on how laning is going, get the upgraded boots either before or after frozen heart. If I'm having to pull back often or am getting ganked a lot, I rush full boots. If I'm keeping the fight fairly level or purely dominating my lane, I rush the frozen heart before the upgraded boots.

I take magic penetration boots to give Ryze that little bit harder kick. Cooldown reduction isn't needed since between his passive, low cooldown times, frozen heart, and your masteries, even without blue buff, his cooldown reduction is pretty nice. get blue buff and you hit max cooldown reduction.

After a frozen heart and full boots, I grab Rod of Ages for the additional health and mana, as well as a little bit of supplementary AP

Finally I'll finish off Archangel's Staff to give that small boost in AP and faster mana regen. Despite Ryze having plenty of mana, if you're spamming enough, you can still burn through it.

Next I give myself survivability with a Warmogs. Between a nice jump in health and some health regen, it's a nice addition for ryze.

I grab a Banshee's Veil if the enemy team has their AP doing any significant damage at this point. If their AP isn't doing much for damage, or if their team is heavily damaging with AD, grab another frozen heart instead.

While Some people prefer to go rune prison first, I find that locking them up to hit them with one basic attack before they run off is kinda pointless. So I grab it second.
I take Overload as a good hard hitting poke to start, and it's spammable. Once I get rune prison at lvl 2, I can dump them both on them right after eachother.
I take Spell Flux at lvl 3, then ignore it till I don't have any other skills to level. Since I'm building towards Mana, it gets no additional bonus (which even if I did build AP, it doesn't get much of a bonus), and I spam it when I can to give tear of the goddess some bonus mana.
I then rush ranking up my Overload and either max it at lvl 8, then max out rune prison after, or max out rune prison starting at lvl 8, and then take Overloads last point after rune prison is maxed.

I grab Clarity and flash or Ignite and Flash
Clarity is fantastic for early game mana sustain so you can really let loose with your spells a bit more to build up tear of the goddess (and save on mana potions)
Later game I spam steal golem buff. But clarity is still fantastic in the event you've died or no blue buff is available.
Last thing you want to do is run out of mana. :P
Ignite and flash is better once you're comfortable with ryze and can get through the occasional early game periods with no mana.

Ryze is starting to make a come back. I've started to see him more often in PVP. He's a viable champion for many roles, and I'm happy to see him being played outside of just mid.

Here's a video playing of me playing as Ryze with this build on my old main (before I created OminousGiggle)
Though my quintessences weren't purchased at this point, and I can't remember what my masteries were, the rest of this guide is accurate to it. (just without enough time to build banshees veil or a second frozen heart)
Ryze Mana Off Tank