I'm no Ryze expert; probably it will take a while for people to really figure out what's best, but I've played him on PBE a bit. Generally you want Tear ASAP, then RoA. You want to finish your Archangel's/Seraph's once you know you'll fully stack your tear before your next back. For a third item (outside of boots, which I generally take Sorc boots or a defensive boot like Mercs or Tabi depending on my needs), I think Righteous Glory is very strong on him, since it gives him more mana and health, but also the MS active which helps him get in range and stick to targets to get his full combos off. You probably want a Void staff, as you do with most AP carries, and maybe something like Zhonyas since your AP ratios are improved, but also new Ryze is less able to build tanky than old Ryze, so the Zhonyas active is really nice.
I've also seen some people get Muramana, since his E procs it multiple times if you attack someone in a minion wave, so it absolutely will destroy a target in the right situation (and will also cost like 1k mana on it's own). It's kind of crazy though, so I wouldn't be surprised if Riot removes that interaction in an upcoming patch.