OK so heres my build for 3v3
SIDENOTE for Noobs: 1 kill is never worth 1 death. I'm always hearing people say, "At least I got them." This phrase is especially true with mejais soulstealer. Don't die. Also, if you are dead then you can't help your team mates at all except by saying gj! or n1!
Summoner's Abilities (Crucial to this build!!!!! Also remember this is for 3v3)
GHOST & CLARITY
getting ghost will allow you to chase low level players running away from your destruction. Using ghost and then throwing a rune prison on them can help your team mates catch up and deal some serious damage. It will also help you run away from enemies. Flash might also be useful here but I prefer Ghost for two reasons. It lasts longer therefore letting you gain more ground and you can buff it by 8% in your summoner tree.
Items in order of purchase
Amplifying Tome and a health pot
Most people try to gank the middle bush in the map. This is plain stupid so if you have team mates that insist on doing so stay behind them. You will be the first to die.. always.
If you got some kills and are a couple levels up from your enemies then get mejais soulstealer
If not then purchase boots and another health pot cause you probably wont be able to afford Sorcerer's Shoes
Since they buffed the Archangel's Staff this is what you should be working for after you get the above items including finishing the sorcerers boots Because Ryze is a carry that means he gets better late game. Don't buy the blasting wand until you have Tear if the Goddess. Once you get Tear of the Goddess use your spells every chance you get on any creeps even one by itself. Everytime you use a spell your mana increases and when you finally do buy the archangel staff you AP will boost significantly.
After you have these items (Sorcerers Boots and Archangel Staff) here is where you decide between two or maybe even three paths.
If you've been getting kills but your team has not been getting very many your going to be targeted and the enemies are going to be buying magic resistance. Get the Abyssal Sceptor because the magic penetration helps your allies do damage with their magic attacks also.
If your dying very often your probably in danger of losing the game or you suck. In this case I suggest very strongly buying the Giant's Belt because this along with your GHOST ability will help you run away if you fail in any small battles. If you do have to purchase a Gian'ts Belt then its inevitable that you should buy Rylai's Crystal Sceptor. Its AP boost is nice but the 35% speed reduction fails for Ryze's best moves Spell Flux and Rune Prison. By time prison is over the 35% speed reduction is gone. and Spell flux is a multi target so really it only slows by 15%. But it is better to finish a build than stop halfway. You can also go for Soul Shroud which gives nice cooldown and mana reduction. Id say Soul Shroud is your best bet.
If your whole game has been going as planned and you have mejai's soulstealer at 20 stacks, you've probably also maxed out your archangel's staff and your team is doing really good then what next? By now you have a huge load of AP. If you want your overload to be hitting in the high 900's then your going to want Zhonya's Ring. It gives you 125 AP and increases your maximum AP by 25%.
160 AP (20 stacks Mejai) + 45 AP (Archangel Staff) + 70 AP (Archangel's Staff 2.5% of your mana to AP assuming you have around 2800 mana by now) + 125 AP (Zhonya's Ring) + 100 AP (Zhonya's Ring 25% of your total AP is converted to more AP) = 400 AP. (I hope the math is correct)
If you have all of this plus you may have some other AP increasing items you are a boss. I think one time I had a base of 512 AP overall which = **********. This doesn't include Ryze's Ult Desperate Power.
These are the basic items you should get for an AP build. After you have all of these items and you still haven't won get Lich Bane. If you can afford anything after this then you should quit because this game has gone on for way too long.
If your enemy is running away or your team mate needs to catch up to the fight then throw down a rune prison. Otherwise get as close to them as you can and use spell flux. After using spell flux throw an Overload or you Rune Prison at them. By now if you used your rune prison it should be back up again because using your spells decreases cooldowns by 1 second. Just throw that on them and they should be dead by now.
Using these moves will be pretty useless unless your using your ULTIMATE!! Your ultimate is great for doing immense damage to an opponent or taking out a whole mob of creeps. Use this spell whenver you can spare it unless you feel you are about to get ganked. Then you won't have it and you'll prob die. Since the cooldown is only 30 seconds and it has 65% AOE damage it mixed with spell flux, overload, and Rune Prison will kill the whole wave of creeps.
OK so everyone already knows that 1 versus 1 your pretty much unbeatable. If you are losing though and you happen to get away then run to the nearest bushes. Lets say you are losing a battle with (insert character here) and you just ran out of mana. You need to get away. First pop your ghost ability and run to the nearest bushes. That (character) will probably chase you. This is a good thing. Use Clarity so that you have mana. As soon as they are about to enter the bush make sure you use your ult if you have it up, if not then u might still win, use rune prison and then spell flux using your body as a bouncing point and finally throw an overload for good measure. You will win because they have no idea whats going on and your nukes will kill them almost instantly.
Same as above but now you can use the two enemies bodies as bouncing point for your spell flux. which equates to twice the damage output.
Kind of hard for Ryze especially if all three are surrounded by creeps. Remember your spell flux jumps to the closet thing to your target. But pretty much the same as above. Use your ult and throw a spell flux. If they chase you use your Rune Prison on the one closest to attacking you and then continue to run.
4v4 and 5v5
Concentrate on one or two guys. Make sure you tell your team who you are going for by pinging that enemy. Go for the Tanks because you can keep them at a distance whereas a range guy can still attack you while held in his Rune Prison due to the newest update.
Well.. thats all I got thanks for reading.
And runes are important but I dont feel like wasting THAT much time so figure that part out yourself