I will explain the items first and why they are good for Ryze. If you have any better thing than two frozen hearts than go on and use it. The frozen hearts just give you time to stun them and ulti them to death while stopping attackers like Tryndamere from killing you. The boots are obviously a good help on your AP boost and speed.
The Lich Bane is for having an AP boost while getting more speed. The Rod of Ages is getting the needed boost for health and mana if you need to turret dive a little.[WARNING DO NOT TURRET DIVE TOO MUCH!] I cannot stress this any bigger. If you die that was your fought not the guides. The Banshee's Veil is for that little thing called survival from a ultimate pawnage! Just if your wondering i mean if you get ulti by Lux she want hurt you. Thornmail is for the bonus armor and the passive of becoming a spikey plated mage.

I dont want to buy 1000 point runes for 2.9 health. Thats it. If you like runes get them but I don't use them like I should.

I always like having a mini starting cooldown or mana boost but masteries are free unlike runes so go knock yourself out with any mastery you like but i suggest these.

Overload is made for that extra bit of cooldown reduction you will need for his moves.
Spell Flux is made for minion farming and clean up for running champions.
Rune Prison will catch your enemy and help you catch your enemies and finish them.

I'm sorry I made it short but it is to the point. If you play right then you can go without dieing. My last game was short and I didn't even finish but I did go 8 and 0. I have never used it in rank but if you make this and have a video using it send it to me and I might put it in later. This will never be completly done and there is always work so I'll keep updating this and for now this is all so have fun.