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 to 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 into the enemy.

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 and mana.
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.

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.
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 (since it's significantly cheaper to build mana than AP) and armor to give him the general tankiness I love so much.

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 grab Rod of Ages for the additional health and mana, as well as a little bit of supplementary AP

I grab a Banshee's Veil if the enemy team has their AP doing any significant damage at this point. If they're not, or their team is primarily focusing with the AD, I grab another Frozen Heart.

Finally I finish off the Archangels staff. By this point I have enough mana for it to give me a significant AP boost (usually 80-120 AP) which gives a nice surprise to your enemies at the sudden jump in damage.

The force of nature is primarily if you're taking on a team with a fair bit more AP. If not, swap it off for a Warmogs to give you the extra bulkyness and health regen. Or, if you've got a good support, go for another frozen heart to give you extra armor dump and mana. If you've taken a second frozen heart instead of the Banshees veil, take a warmogs.

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 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, 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 ignite and Clarity or Ignite and Flash
Clarity is fantastic for early game mana sustain so you can really let loose with your spells a bit more (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.