I've been playing LoL for a while now and i have recently started playing as Ryze and i particularly like this build especially late game since no one is able to stop your magic penetration and attacks. Ryze is particularly good at team fights and is able to gank players easily with his rune prison ability, providing he is with someone else. But late game with these items he is able to take people on in one on ones easily.

I chose this item order for a specific reason.

Sapphire Crystal - This is standard with any Ryze build as the increase in mana is needed at beginning of the game, because ryze is very mana hungry at the beginning of the game, also it adds an attack bonus onto Ryze as you know.

Tear of the goddess - I chose this to come second as you find out you will be running out of mana very quickly with ryze so you want to get more and with tear's bonus mana you want it as soon as possible.

Boots - Standard boots, you want to run faster and then you upgrade.

Sorcerers Boots - Standard choice for Ryze as magic penetration is with any mage. Makes your attacks more powerful.

Archangel's staff - adds that much needed AP to make your attacks stronger and more deadly and it continues that bonus mana from tears of a goddess. Once you have got this you should be able to do some damage and pick up a good couple of kills and assists.

Void Staff or Will of the ancients (optional) - This choice is up to you which one comes first but i usually go for void since it allows me to do more damage and you wont need the health if you play smart and hunt in a pack. Then i get will of the ancients which gives me this spell vamp bonus to me and my teammates. You are usually really powerful at this point as there isnt much the enemy can do to stop your attacks and if you play smart with your team you can easily get some kills.

Frozen Heart - is more of a item you dont usually get to and is more of an item to annoy your enemies as it slows them down and simple makes you more stronger but it isnt all much use early game (in my opinion) so i choose to get it later

Banshee's Veil - Another item you dont usually get to use since you get it so late but is great in a team fight and simply makes your attacks more powerful with that mana bonus and you will need it at times.