BEFORE ANYONE POSTS BAD COMMENTS TO REFUTE THIS BUILD BASED ON AP+MANA
Be aware that I am not pro at this game, nor do I wish to waste my life getting the exact details of the best considered build. I created this build because it works decently in both 5v5 and 3v3. I do not judge your intelligence for what you think you should use, so do not judge mine. It works for me, and that is all you need to know.

I welcome your criticism, but do not do it in a manner where you would be trying to describe me as something you would appear to be.

I understand that Ryze is unique because his abilities get buffed based on his maximum mana, but I believe that if you focus on increasing your maximum mana at the same time as increasing your ability power, together with magic penetration people run away in fear from Ryze's AoE attacks and capability to spam a high damaging overload.

Before rating this build, please try it out at least. Focus on spamming overload as soon as it refreshes when you have acquired the 'Tear of the Goddess,' makes it easier to last hit minions too so getting the rest of the build afterward should be no hassle.

Here is a picture of the stats at end game, these include the 1000 mana bonus the 'Archangel's Staff' gives:

[WORKING ON IT]

These masteries do several things that are useful for playing as Ryze.

First off, the maximum mana and mana regeneration are both increased, further boosting Ryze in what he does best, and allowing him to lane longer without mana deprivation.
Ability power gain per level is also boosted, further increasing Ryze's damage.
Ryze's abilities have their cooldown reduced (9% from masteries), and pierce the target's magic resistance.
Exhaust keeps opponents close and lowers their resistances.
Ryze levels up faster, and if he does happen to die, he will spawn 3.33% faster than the normal respawn timer.
Lastly, Ryze moves 3% faster to get where he is needed quicker.

This item sequence is recommended if you want this build.

However, although I put a health potion, I suggest you buy a mana potion if you do not planning on ganking in the early game or fighting over the bushes in your lane.
Wards are also recommended 5 minutes after the game has started so that you can be prepared for a gank and not get your item collection on halt from the respawn timer.

Also the 'Banshee's Veil' can be replaced with 'Frozen Heart', if you do not need the magic resistance. The frozen heart grants more mana, and therfore more damage with abilities, not only that but it also further increases your armor making you harder to kill from physical attacks and further reduces your cool-downs allowing you to hurt more in less time.

This skill sequence focuses on distributing the mana cost of each ability so that you can remain in your lane for as long as possible before having to go back due to mana deprivation.

Depending on your play style, and that of your enemies, feel free to change it up, but the sooner you can get your ultimate to level three, the sooner it will refresh for you to use it again.

If you wish to farm, keep overload at level one until you need it to do more damage, and just use it for last hitting minions and permanently increasing your maximum mana by 4 as soon as you get the 'Tear of the Goddess'.

If you need to hold enemies, then try and get rune prison to level five as soon as possible, although it will cost more mana, it will keep your prey in the prison longer and deal more damage to begin with.

Exhaust and Flash are the two summoner spells I chose because these are the most useful for my play style.
Exhaust to prevent opponents from running away and healing, at the same time as reducing their resistances with the masteries provided.
Flash to flash away behind walls when I need to get away... you know, when things don't go as you planned.

Feel free to change these up... I find ghost also useful for catching up, and clairvoyance for scouting.
But two spells that I don't recommend are heal and clarity. Heal is just noobish because if you play Ryze, you shouldn't need heal. Clarity may be useful for you to burst out damage several times if you can time using it properly, but thats only at the beginning of the game, late game it's useless because you will have too much of it anyway.

As Ryze in a group you have the option to either initiate a team battle with your ultimate and spell flux, keep key opponents from running off, or providing an escape route to those who need to get away from opponents.
Either way, focus on providing an advantage to your teammates, because in the end if your opponents feed on your allies, even you won't be able to hold them back from taking your nexus.

Ryze is beast if you know how to play him properly, hopefully this guide helps you a bit in learning how to play him, if not, then you already know how to play him and you don't need my advice.

Also keep in mind. If you happen to die in the beginning more than you'd like. Don't grief, keep your cool, you can catch up. The item set is relatively cheap, you can still recover from your losses. No one likes griefing, ruins games and benefits no one. Not to mention it is also against the player code of conduct, and you will get reported for such behavior.

Go and lead your team to victory, and don't forget to rate and leave some some feedback!