I generally don't go soulstealer as one of my first items unless I'm confident I have an advantage and can keep it. I typically go deathcap first instead. Also, I personally am not fond of Rylai's on Lux as much, though if your team needs the CC then it's not bad. I typically go for Voidstaff instead which is cheaper, gives near the same amount of AP and gives you 40% more magic penetration. Also, it probably wouldn't hurt to get a defensive item such as hourglass or banshee's. Depends from game to game but getting a scepter late game for the magic resistance isn't going to help a great deal vs an AD carry. If you're going for pure AP then you should definitely consider Hourglass as it does give 100 AP. combined with deathcap and a few other AP items you can get your AP well over 600 (especially if you got your stacks from soulstealer up to 20).

I personally think you should redo your runes as AP marks and seals are not optimal. They give very little benefit outside of the early game and you'd be better off getting some magic penetration (excellent in early-mid-late game). Mana regen seals are also a good choice.

I would also recommend going 9/0/21. Getting Archaic knowledge and filling up utility. Be sure to grab the 2 points for the blue buff. You'll want the blue buff to keep the mana regen up and your cooldown on your ult down to 24 seconds.