With the recent overhaul to everyone's favorite purple faceroll mage, the necessity to build Ryze strictly as an Ability Power-based caster is simply defunct. Ability power scales horribly with all of your abilities, and in its place Overload and Rune Prison both scale in damage based on your maximum Mana.

What this means build-wise is that Ryze wants Mana, obviously. What this means for his skill sequence is entirely different: Ryze is no longer best as a single-target Hiroshima re-inaction, and is now played best as a DPS-style attacker.

What's the immediate benefit?

By building Manamune first, Ryze can not only build up a load of Mana to pump up his spellpower, but he also gains something that he didn't have before: lack of reliance on his cooldowns.

For boots, I prefer to use Ionian Boots of Lucidity for the cooldown reduction. In combination with your runes, passive 10% from Overload and Masteries, you should be high enough with CDR to not warrant useage of other Cooldown Reduction items, and you can always top yourself off by taking the Golem buff, or simply popping an Elixir of Brilliance

Archangel's Staff is taken after this in order to further increase your mana pool, but also because having both passives from Archangel's Staff and Manamune allows them to double-dip into one-another. What this means is an immediate - and noticeable - increase in your Q and E damage, physical damage and ability power.

(But I thought Ability Power was bad for Ryze now?)

Too true, but that's not the point. The additional ability power is of little to no consequence; it's just a cherry on the sundae. What WILL benefit from this, however, is building Lichbane

What Lichbane does for this build is incredible. It gives you more Mana, which you ALWAYS want, it gives you more Magic Armor - which will help squishy Ryze live to fight another day, it gives you a bit of Ability Power, and it gives you more mobility to help chase or run away. The very best part, however, is the passive.

(Passive: 100% chance when an ability is used that your next physical attack deals an additional 100% of your Ability Power in damage. This effect has a 3 second cooldown.)
Since Ryze wants to be casting his spells whenever possible, this just adds even more *umph* to your already beefy physical damage thanks to your ungodly mana pool plus Manamune.

The very last item that benefits the build is Hextech Gunblade. What this does for the build is lovely for when you're feeling extra-fed, and desire a bit more survivability and damage at the same time. You gain spell vamp, which will let you heal via your ridiculously strong spells, and your physical attacks will also heal you for a bit as well. This makes you incredibly hard to kill, and incredibly hard to survive an assault from. The active ability is also lovely to have against enemies that move quickly, and can escape your onslaught after your Rune Prison has expired.

IN CASE OF EMERGENCY

If you find yourself in a game against champions who have an arsenal of powerful disables, you can always swap out Ionian Boots of Lucidity for Mercury Treads and swap out Hextech Gunblade for Nashor's Tooth. You may have to be more liberal with the Elixir of Brilliance useage due to your neglected CDR, however, but once you have your Tooth, you will actually be over the CDR cap by a tad without an elixir.

At any point, a Banshee's Veil may be built if needed. The Additional survivability aside, Banshee's Veil gives a good amount of Mana to us as well, which factors into having more physical and ability damage also. Never a bad choice, but there are better things to build if you feel that you don't need it to live.

The Long and Short of It

PROS:

Sustainable damage output
Not nearly as much reliance on cooldowns
Ability to poke in a lane without wasting mana
Viability against champions with high Magic Resistance
Good for unloading against multiple champions at once in a fight
GREAT at ripping down a single target
Physical damage output means you don't hit towers like a girl anymore
Able to easily carry games

CONS:

Very squishy
You are likely to be a very high-priority target
Susceptible to CC and disables

The 1v1 Nuclear Payload:

Overload > Rune Prison > Spell Flux > Overload > Physical attack > Overload

Without using your ultimate, you will still leave a pile of ashes where your target was standing. With full CDR, this combination can be pulled off within the span of a few seconds.

Carpet-bombing a Team Fight:

Overload > Desperate Power > Spell Flux > Overload > Rune Prison > Overload > Physical Attack > Overload

This high-power combo will dish out massive Area of Effect damage, as well as completely obliterate one target. The reason I use Spell Flux after Desperate Power but not before Rune Prison is for the added bonus of lowering the Magic Resistance of several targets, which equates to you dealing much more damage with your AoE from your ultimate while you tear someone a new one.

Rune Prison usually gets used in the middle of the payload because in teamfights, the enemies will try to kill you too. They won't back off 'til they notice they're taking too much damage, and after two Overloads and the direct hit from your Lichbane proc, they're gonna want to start running. Politely deny them the ability to, and then finish them off.

Playstyle, and How this Build Affects It

Normally, Ryze is the type of champion that wants to wait around for all of his cooldowns to be up so he can run in and nuke the living hell out of someone, and once that's over with he runs away. He's a sneaky bastard that likes to roam and gank for this reason.

Building a sustainable damage-dealing build such as this allows Ryze to stand his ground no matter what the situation, and he can defend or push a lane very effectively. In games where the enemies have multiple wards up, or are very mobile themselves, Ryze simply can't afford to try to gank a lane, only to have that ability denied to him. With this playstyle, Ryze can easily farm the items and enemy champions in his own lane while protecting your precious towers.

The Hybrid Ryze playstyle is not very much different from the Ryze we all know and love, it just simply improves him in areas in which he was heavily lacking before without sacrificing any of the things that made him good in the first place.

All I can say is, try it!

UNDER CONSRUCTION!

As this is my very first guide, I am constantly tweaking things I have noticed that need to be changed. ANY helpful criticism/comments will be greatly appreciated. Thank you for reading!