Yet another victim to the inability to do calculations.
Here, let me do the calculations for you. With an Archangel's Staff and Rabadon's Deathcap, you'll most likely hit 300 AP (or thereabouts). With 300 AP, you just gave yourself 105 AD with your Q, and 60% attack speed. You have the effects of nearly FIVE items in two. That's nowhere near balanced. Riot's been trying to cut down on double-dip passives that increase another stat for increasing one stat. It's not a good idea at all, especially since you can build AP and get everything. I'm not sure what you were thinking.
Irredeemably broken double-dip min-maxing passives aside...
The Q is useless. Changing your attacks to magic damage isn't going to do anything of note - in fact, it means Morgana can now block your hits with her shield. If anything, it's a nerf, especially with the no-crit clause.
The W is even MORE broken. Come on, man, you're killing me here. What is making you think that tacking on AP ratios to buffs is a good idea? No champion has that, and for good reason - AP hits absurdly high numbers! With that 300 AP, you can get 120 more damage on autoattacks, 150 life healed on autoattacks (more sustain than Warwick and Irelia combined - see a problem here?), or 60% more attack speed (you'll end up overkilling attack speed if you keep building straight-up AP). All this on a cooldown that can be reduced to 7.2 seconds, ensuring perma-buff. No. Just, no.
For a gap-closer, that does a metric s***ton of damage. It doesn't need added effects, either.
Oh, COME ON. Let me crunch some level 18 numbers for you - assuming you have 100 base AD (not unreasonable for a melee DPS champ - UP, actually) and 300 AP, and pop your (immensely broken) ultimate, you have 100 + 105 (from the Q) + 120 (from the W) + 100 (from the R) = 425 AD. 425. Tryndamere and Yi won't hit 425 AD. You can do so with two items. I didn't even bother adding in the 100 AP to all the other crud you've got going on, either, so the end result is EVEN HIGHER.
This is not the way to make a steady-damage caster. You centered this champion entirely around double-dip passives and autoattack steroids. It is inherently broken - ditch this champion, think about some game balance, and try again.