Ryze recently was given a remake by Riot, and now he's not only viable, but he's TONS of fun. Seriously, I literally NEVER played Ryze before, but now it's at least every other game. Anyways, here's my advice and take on playing Ryze in mid-high elo, hope this isn't too bad, as it's my first guide.
-Strong early game, able to take someones hp down to half with one proper combo.
-Ability to carry as a caster
-Able to be durable, and put out a great amount of damage at the same time.
-Does well against the "tanky dps" meta.
-Good attack animation for last hitting.
-Can't carry as well late game as someone as say, ashe.
-Mana reliant early game, mp5 runes are required.
-No true escape, why you take ghost and flash.
-Solo lane is required, reliant on level scaling.
-Somewhat waste of an ult.
I generally start out with a mana crystal and 2 hp pots. Since Ryze was remade you basically just need mana items, Ap now scales terribly with Ryze. On your first trip back you should have enough for ToG and boots of speed. You want tear asap so that you can begin charging its passive. Second trip back you'll want to have enough for catalyst and mercury treads. I prefer Mercury Treads because Ryze is very vulnerable to focus fire, and sorc boots can be somewhat made yp for by your E's debuff. Next build a Frozen Heart, then finish your Banshee's Veil. After this, it will be nearing end game. Build a RoA, and void staff for the Mpen, and if the game is somehow still going finish your AA staff.
As Ryze you're going to have a solo lane. Ask your team in the select if you can have one. If not, play somebody else. This is mainly because Ryze scales so well off of levels, and if he's getting out farmed and out leveled by other carries he's obviously not going to preform his role as well. I really like to solo top because it's isolated from the rest of the map, so you'll be able to free farm for a long time there. Mid is way easier to gank since you can from either side lane, but it doesn't really matter. up to level 3 last hit with auto-attacks and don't push your lane, you have no escape mechanism. Harass with Q if your enemy gets in range, because Q has a longer range than most champions auto-attacks. If you're against a melee champion, take advantage of the brush in top and zone him as much as possible. If they come up for a last hit punish him by giving him a nice Q and autoattack, but run away after to drop minion aggro. Learning your spell combo is a big part of learning Ryze. Your general combo is Q-W-E-Q. This should bring your opponent down to about half hp in one combo. The reason I use this combo is because of Ryze's passive. I like to use Q first then snare the enemy, then after you get your E off not only will your Q already be up again B/C of his passive, but your enemy's magic resist will be lowered from Spell Flux. If you've got an opening to land a kill, flash in and pick it up, summoners are worth using to get kills. if you're against someone like vlad, irelia, urgot, or sion just try to farm and not get denied. If your jungle comes for a gank check your opponents summoners and if they have ghost flash don't waste your ghost, just scare him into blowing both of them. If they only have one, or neither, pop ghost and USE YOUR SNARE FIRST so that you can get it up again with your passive asap. If you have ToG use Q and W to last hit creeps to charge its passive. Feel free to waste your ult, since it's not very good and easly refreshed (low cooldown plus your passive.) If you're top and the other team 5 mans dragon or mid turret and there's no way you can get there in time just push top as hard as you can and try to get the tower down. Blue buff isn't required on Ryze, as you'll have a huge mana pool anyways (4k+), but it's really helpfull.
Wait a little to run in (initial aoes, stuns, mumu ults, etc...) then run in and combo the nearest squishy, but don't put yourself in a bad position just to get to their carry. keep using the combo I mentioned above to get your Q up and make sure you use your ult early to take advantage of the aoe and spell vamp. You're not likely their number one focus, mainly because you don't do as much damage as an AD carry, and you're way more tanky, but if you get caught in a mumu ult, or a malphite ult make sure you flash out. When chasing stay in range for a W then unload a spell combo, when retreating snare the biggest CC threat, ex: nunu, udyr, nasus, and if your carry starts getting focused make sure you do your best to peel. By end game two Q's, a W and an E will kill somebody like Ashe. Remember, Banshees veils on the other team is your enemy. Try to Q off the veils, since the refresh time was recently increased. Don't initiate a fight unless your team has a really obvious advantage, in that case flash and snare an out-of-position squishy.
Ryze is one strong mage carry when played properly, and I've seen so many bad ones that it's beginning to make me sad face, so I hope you guys learned from this guide, if so, give me some feedback, if not, give me some negative feedback!