Ryze - The Rogue Mage

He is one of the few mage carries in League of Legends - all he needs is a lot of AP and mana to dish out some damage.

Pros:
1. Consistantly powerful damage.
2. Needs only Mana to be successful.
3. Rune Prison - A powerful nuke as well as a Snare.
4. Spell Flux - Magic Resistance reduction.
5. Desperate Power - Bread & Butter Spell with amazingly low cooldown.

Cons:
1. Easily killed.
2. First Targeted in Ganks.
3. Useless if Disoriented.
4. Useless if Silenced.
5. Useless if Stunned.

[Intro taken from Alahric's "Ryze - The Glass Cannon" build... too lazy to write an intro xD)

ITEM SEQUENCE

Sapphire Crystal and Health Potion starts off the game quite nicely. After that, get Sorcerer's Shoes or Ionian Boots of Lucidity, whichever is more useful. If necessary, Mercury Treads provides some decent Magic Resistance and CC Reduction.

Catalyst the Protector is a decent item to go with next, though you could go for a Tear of the Goddess, for more Mana. Why Catalyst? Because it builds into 2 great items for Ryze: Banshee's Veil and Rod of Ages.

After Catalyst, get Tear of the Goddess. This item is self-explanatory.

After Tear, Rod of Ages is next. You probably still haven't maxed your Tear out yet, and RoA provides a nice boost to Mana and AP as well as an icnrease in Health.

The next item on our recipe is Frozen Heart. Now, this part is interchangeable - you can either get Heart, Banshee or Archangel - either Heart of Banshee work, depending on your enemy team's composition. I wouldn't recommend Archangel yet though, since your Tear probably still isn't maxed yet.

TIME FOR ARCHANGEL. If you still haven't maxed out your Tear, just keep farming. Otherwise, upgrade it and start spamming some more.

Now that you have Archangel, get Banshee's Veil (or Frozen Heart, whichever you didn't take first). After getting it, you should be dishing out lethal amounts of Magic Damage whilst upholding lots of survivability.

Now here comes the fun part. Games don't usually last this long, but if they do, then Ryze needs that extra push - I usually go for Lich Bane: Mana, AP, Magic Resistance and Movement Speed, all of which synergize extremely well with Ryze. Who doesn't like that? And after using a spell, your next attack deals 100% of your AP in damage. ONE HUNDRED PERCENT OF YOUR AP.

If you don't like Lich Bane, Rylai's Crystal Scepter is a good choice - it provides a nice slow, though I wouldn't recommend it as it doesn't give anything other than AP.

If not Rylai or Lich, then get another Archangel's Staff. It gives a nice dose of AP, and more Mana is simply helpful. Or you could go for Rabadon's Deathcap - a HUGE dose of Ability Power, though it isn't really necessary.

Anything that gives mana/AP/CDR is a viable item. Items like Deathfire Grasp (AP/CDR/Nuke) are viable choices as well, though DFG isn't really as recommended anymore.

EARLY GAME

Sapphire Crystal is by far one of the best items to get, at first, for Ryze. The Mana boost is excellent for Overload and Rune Prison. Getting Doran's Ring isn't worthwhile since, if you're following this guide, you're not getting Spell Flux first (which is a ridiculous choice, I believe, since it doesn't do much damage anymore).

Whether you're solo-ing mid, or laning with a partner, harrassing with Overload is is your best bet. It has gained a range increase since the Ryze overhaul, and its extremely low cooldown is perfect for burning down Health. Mastering these tricks will give you a huge lane control advantage. If used correctly, they will FEAR you and be zoned out often. Flash coupled with Ghost provide both chasing and escaping mechanisms, so don't be afraid to spam that Overload!

MID GAME

By now, you should have Desperate Power, minimally Catalyst the Protector and/or Tear of the Goddess for Mana. If you haven't been able to get any kills yet, now's your chance. After handling a lane for a little while, go MIA and start moving towards any squishy target. If there's a mid Champ that hasn't been farming too much, they're easy targets. Set up the kill with your mid partner and it's in the bag. If not mid, then support top - Rune prison is an amazing Nuke, and coupled with Overload, massive amounts of damage can be dealt. Or assassinate if they're overextended. If you're unable to get kills, keep farming and occasionally support your team until you get Rod of Ages - you'll get your chances soon enough.

LATE GAME

If you haven't been getting many kills, I suggest you start following your tank around and help set up ganks. Your constant DPS Magic Damage will help a LOT in many a team fight. If you've been well fed, you can risk occasionnally venturing out and assassinating the stray backdoorer. With increased survivability, you can snipe and try to outdamage carries, since Ryze is probably one of the game's only DPS Lategame Carry Mages.

SKILL SEQUENCE

Try not to brushcheck early game. Your damage is quite low, even if Overload's cooldown is nearly nothing. You're much squishier as well, without Doran's Ring. If you're able to enter Brush before your opponent, then prepare for a surprise attack - especially if you took Rune Prison instead of Overload first.

As the game goes on, you'll start by nuking with Overload, following up with a couple of bounces from Spell Flux, Overload again, Rune Prison, Overload, etc. It is important to max Overload first - you get more passive Cooldown Reduction from it, which synergizes greatly with Ryze's passive and Ultimate (now grants a passive boost to Mana).

Once you start Assassinating, be sure to use Overload first. Once you use Spell Flux, Rune Prison or Desperate Power, your Overload cooldown should have been reset already. Using your skills in quick succession allows for a rapid kill - start off with Overload, then Desperate power, Overload, Rune Prison, Overload, Spell Flux, Overload. If the laners you're helping do well, a double kill is in the bag. If you're helping Mid, a nicely placed Stun, Supression or Immobilizer will ensure a quick kill - if not, a lot of damage.

*NOTE

Switching out Ghost or Flash for Ignite is a decent idea as well. Getting that last bit of Health that you were unable to get with your regular attacks is always satisfactory. Also, if you get plenty of kills, you can start getting cocky. Your opponents will hate it when you say "BZZT" each time you get a kill (make sure you're winning first though, or you're just gonna seem ridiculous).

And that, as they say, is that.

Following this guide won't necessarily make you the best Ryze out there. It all depends on your skill as a player. However, if you play well, your oppoenents will definitely fear you as they would Eternal Thunder.