I haven't reached much ranked yet, I am a bronze 1 player with 12 games.
In unranked games I would estimate that I won "most" of my Ryze games ~70%, after I learned how to Teleport better, I chose it over Ignite. Ignite wins you the most 1v1s in your lane, not only with Ryze, but he is able to farm and trade properly at the same time against the most opponents, he gets more kills in general when he teleports to a bot lane ward.

Spirit Visage and Frozen Heart are very similar items on Ryze. They both give the same amount of cooldown reduction and a lot of resistance. With one of these items and the utility mastery tree Ryze gets 40% cooldown reduction since his Overload already gives 10%.
Spirit Visage gives huge magic resistance and a passive that works with Ryze's Desperate Power while Frozen Heart gives enough armor for your whole build, damage through mana and an active against ad carries.

In most of my games I choose to buy two Rod of Ages just because of the massive amount of health. A Warmog's Armor gives 350 more health but you lose a lot of damage. A late Rod of Ages isn't as good as an early one, but it still gives 450 health and mana which is huge.

Seraph's Embrace/ Archangel's Staff is again the only possible upgrade of the Tear of the Goddess after Muramana's damage became physical and because of this it doesn't fit to magic penetration.

Abyssal Mask is a good item for hybrid resistance (with Frozen Heart). With Spirit Visage and Abyssal Mask you will mostly not be tankier than with a Frozen Heart. Abyssal Mask gives an additional damage against carries and squishies which makes it one of the best bruiser items.

Ryze is not only a bruiser, Frozen Heart and Rod of Ages make you as tanky as your fulltank jungler, since he gets less farm than you and you get your items quicker than him. In a teamfight your want to position yourself in the middle of your enemies and burst everything with your Desperate Power so that your ad carry can shoot from behind without trouble.

Teleport gives you mobility, which is all that Ryze needs to put damage down since his range nerfs. The best way to have success with Teleport is to tell your support to put a ward in the lane brush. You should Teleport to this ward when the enemy bot lane has pushed, otherwise they will run away from you. Top lane ganks also work, but top laners rarely ward the lane brushes so it is harder to get behind the enemy top laner.

You want to level up Overload first because of its low cooldown, its passive cooldown reduction and the ability to trade damage in lane. I really love the extra magic penetration from Spell Flux. Also the extra damage will help you clearing a fast wraith camp. The advantage of leveling Rune Prison secondly is the snare duration increase by one second which can help out at ganks.

Many people use the 9/0/21 mastery tree for Ryze. The main reason why I choose it is because of the 6% cooldown reduction and movement speed. Mana, summoner spell cooldown reduction, increased buff duration and increased experience are cool stats. You can go for 9/21/0, it makes you a lot tankier, in this case you can go for cooldown reduction quintessences. The Pickpocket mastery gives you a lot of gold because Ryze always auto attacks after using spells. The 1 minute ward is good against early junglers like shaco or in situations you feel uncomfortable

Ryze has a lot of trading power. You mostly want to harass your opponent with Overload and 1 auto attack. From level 2 you can Q your enemy, snare him after that and auto attack him once, then run away while he's snared. At level 3 you will unleash your real potential. Basically at level 3 Ryze has reached his strongest point in the game. As long as you have mana you can spam your opponent with huge damage which mostly takes 50% or more of their health. Your passive Arcane Mastery makes this possible. One of the best combos is the Q-W-E-Q combo which takes 4 seconds in total. Once your target is snared it can't escape from the last two spells. Another good combo is the Q-E-W-Q combo. It deals more damage because the magic resistance gets reduced earlier by Spell Flux (this effect lasts 5 seconds!) so that W and the second Q will deal more damage. I like the second combo more but it's riskier, you want to use it when you're very close to your enemy, i.e. when flashing in front of him.

Against Baron/Dragon you use the Q-W-Q-E-Q-R-Q-W-Q-E-Q combo which you can spam without waiting for cooldowns. Ryze can solo the Dragon very early, let's say after you finished your Rod of Ages. Ryze can also solo Baron Nashor with only a few items, Rod of Ages, Seraph's Embrace and Frozen Heart reach out. This video shows how it's done Play Video

Ryze has received an incredible nerf with the range nerfs, losing 25 range on Q, 50 on W and 75 range on his E, this makes him less viable, he has one of the lowest win rates on 5v5 at the moment after champions like Olaf and LeBlanc. He has the 3rd lowest win rate (44,36%) in all ranked and the 5th lowest win rate in Diamond league (45,37%). He also has the 9th lowest win rate on 3v3 (43,79%)
The reason why Ryze is so weak right now is because he can't chase people anymore, people still take movespeed quints on him though, it's much better to take mana quints on Ryze at the moment, this guide is a little outdated. He also can't trade in lane (this is so important because what made Ryze strong was his dominance in lane and win 1v1s). He has been played in the S3 world championship 2 times by Ozone because he countered Ahri in lane (before the nerfs) and he lost his lane both times because of 1v1s, he also hadn't a big impact on the game, sad to watch.