Anagram, Ryze is a good teamfight champ in that he basically takes one other champ out of the fight straight up. He's not an AP carry in the sense that he's literally the worst pusher in the entire game. He makes up for it in nuking potential with his passive/Q/W being overpowered.
go with 9/0/21 and w/e runes you want and flash/ignite
start mana crystal and 2 pots
go back when you can afford tear of the goddess
buy boots whenever
catalyst
RoA if you want W/E to do more damage, but I prefer banshees
frozen heart helps pretty much all around
at this point you should have nearly 3k mana, a decent bit of hp, good resists. the build up to this point is his core and are the items I take in 90% of my games (tear/catalyst in 100%)
next buy w/e you need like magic pen, more health, more mana, AP if you want to be a better pusher, etc
his combo is IDEALLY Q>W>E>Q>R>E>Q>W (wait a second) Q
but this assumes you are within range of Q, and don't need to hold them down first
if they're out of range of Q and within range of W, go W>Q>E>R>Q>W (wait a sec) Q
basically anything to blast as many Qs as possible because they're worth 600 damage apiece
pick up his ult at level 6/17/18 (opting for W at 11 and E at 16) and use it for farming or healing up between battles. Other than that its only use is to make your Q come up sooner in the combo and it's not actually worth taking level 2 R over maxing W or E sooner.
I have some more tips but this post is running long, let me know if you want to know more