Well, this is my first "guide." I think I should say it's more of a "this is what you get" guide, not "this is how you play Veigar" guide.
I made this guide because I've tried high-AP builds, and they never work. I'll get decimated by about any champion if they know how Veigar works... Very painful. I can't farm my Q if I'm dying under my turret.
I tried out a joke set of items, and, er, it worked. And worked the next match. And the next. I only once had any trouble, and that's because I didn't know how Twitch kept making me blow up under my turret. o_o
Anyway, this isn't a super-powered Veigar guide. Just a guide to make yourself be able to kill people, but not. Die. So. Much.

【】 •【】 •••••••••【 •】 ••【••】 •

This is usually how I do things. A • means "separate trip back," things 【inside】 means it's optional on if you want to do what I suggested, and for potions, how many. Lots of potions early on help LOTS (I even buy 3 of both when I buy my Tear)... And sometimes I feel like I want to B earlier than planned to get extra health/mana, if I feel I have trouble with either of them.
I'm not used to this site's coding, so I'll have to make the item order cleaner later. Need to figure out how to make the •' and 【】's more visible.
I didn't list it, but Wards help when you have the chance to be ganked, or the enemy hides in bushes, or you're fighting a disappearing champion.
About Abyssal Scepter. Your sixth item can vary, depending on what your match-up is. Thornmail, Rabadon's, Force of Nature, Mejai's Soulstealer, heck, even Hextech Gunblade will also work. By this point you have all the essentials. I just chose Abyssal Scepter because not only does it grant you more Magic Resistance, it also lowers the MPen of your enemies. Haunting Guise also sounds like it would work, if there are no AP enemies to worry about... And I guess Void Staff would work if there is no threat to you at all, though I have not tested its effect.

Skills:
I level up my R quickly, for obvious reasons.
Then get Q because, well, this is the reason why you don't need that much AP. You can get as much AP as you want as long as you farm carefully.
Then I get E instead of W because I find its stun more useful than the nuke. Veigar needs a stun to escape ganks, stop the enemy from attacking/running away while attacking, and it's useful for ganks if you kill the person in middle without a scratch. Also, your W doesn't work well without your E's long stun.
Last, W. By this point you'll have a Veigar-huge amount of AP (unless you had trouble farming), so your nuke should help you kill people, and push lanes if you don't want to farm your Q at the moment, want to push the lane fast, defend your lane and make enemies back off, or just want tons of gold FAST. Or just for the heck of it. BOOM!
Please not that I'm not sure if it's possible to get the skills as fast as I've put them in the skill order, I've never paid much attention to exactly when I can get moves. Just go Q, E, W, max R/Q/E/W.

Items:
Rushing Tear is perfect for early-game-mana-starved Veigar.
Then Sorcerer's Shoes because you'll want MPen at some point, best get it on your shoes for convenience's sake. You don't know if you'll want to build your 5th item into a MPen item or not, best be safe.
Amplifying Tome is pretty cheap, so I get it fast to make farming easier. But then I get Giant's Belt instead of Blasting Wand because you'll really want the massive amount of Health. Instead of like, 500 health, you have 1,000. Far harder to kill. And you can stay in lane longer, not to mention be less scared to farm your Q. Then you can get your Blasting Wand.
Why the DONUTS do I get Rylai's? Well, I thought about getting Rod of Ages instead. And I've tried rushing Rabadon's or Mejai's Soulstealer... Always ended badly (sorry, Warlemming's guide, you're a dangerous guide.)... It only works good if the enemy is careless, and allows me to nuke them and get fed. But a guide relying on you getting fed is bad, right? D:
Anyway, Rylai's gives you health AND AP. Not to mention a slightly helpful slow (just cast your Q on someone and then you're sure to land your E, also helps if you're running from someone). Why not Rod of Ages? It's cheaper, yes. Yes, it gives mana. But I like the 20 AP right away, I don't want to have to wait for it to be honest. And it lacks the slow. But I guess you can go for Rod of Ages if you want. ^^ It works too.
I get Chainmail as my first item for Glacial Shroud, and then (usually) skip Sapphire Crystal... I find AD gets me killed most of the time, and I don't have terrible mana at this point, so I just buy Sapphire Crystal and Glacial Shroud in one go to save time. Same reason I get Frozen Heart plus the Cloth Armor in one go. With Frozen Heart plus your E, you'll live most AD attacks. Hopefully. And the cooldown reduction is lovely!
Catalyst for more health, and it's also easier to buy it first than Negatron Cloak. Negatron and Banshee's Veil in one go... Though, if there's a problem like a fed LeBlanc or you're facing another Veigar middle, you can get this earlier. (Note, if you're fighting Veigar middle, get Banshee's before Frozen Heart.) Banshee's will give you the Magic Resist you need, and blocks one negative spell! See ya, Blitzcrank. Nice arrow, Ashe, that tickled.
Blasting Wand again because it's the more expensive item, then Abyssal Scepter. Remember that this is an optional item you can switch around when the situation calls.
Then finish off your Archangel's, because you'll have TONS of mana by now, and why would you NOT finish it?
And remember guys, potions and wards, they're great! Except for mid and late game, then potions suck. =(

Masteries, Runes and Summoner Spells:
Eh, I guess they just look reasonable...? *slapped across the face
With the Masteries, Runes and Frozen Heart, you'll (if my calculations are correct, please be so!) have full Cooldown Reduction at level 18.
I prefer instantly having lots of mana and MPen, as mana won't be as much a problem late game, and MPen will also be harder to get due to you might want to switch your 5th item for something else.
Teleport, because you'll want to go back a lot (90% of the time I only B when I have Teleport up), and Flash is just the ultimate summoner spell evar.
Running Quints are awesome, ok? It's ALWAYS helpful to run faster. It's the safest Quints to take. Imagine being able to dodge most skill shots at level 1. Cait is no longer a problem.

If I get any really good matches with this build, I'll put them here. :V I often forget to save good scores, grr.

Thank you all for reading and maybe trying this build!

I will try to clean up this guide later, making links and colors and all that nice stuff to make this less of an eye sore and more of a piece of eye candy (except for the "why" part, that'll always look terrible!)