Now that I'm no longer typing on a phone, I can give my full rundown on Veigar.
First things first, let me just get the Runes and Masteries out of the way first.
In text form, MPen Reds, Flat AP Quints, Scaling AP Blues, and either MP5, armor, or HP/lvl yellows. 21/0/9 masteries, taking AP options in offensive, and MS in utility. Havoc and Good Hands are purely preference, you can put the points into 4 Mental Force and 2 Butcher, and 3 in Expanded Mind if you wish.
Next easiest to go into detail on is itemization. I'm going to say right off the bat, Rabadon's and Void Staff are non-optional. Rabadon's gives +30% AP, which is insanely amazing and I shouldn't have to explain why. Void Staff, on the other hand, is the unsung hero of APs everywhere. It means that at least half of your damage is completely unmitigated. MR is no longer a hard counter. Mkay? Mkay. So, 2 items down.
Next up, some form of upgraded boots are required. There aren't any options in the entire game that let you keep that same mobility except for 3 PDs, which is absolute shit. Of the upgrades, the ones that Veigar can make use of are Swift, Mobi, Tabi, Mercs, Sorcs, and CDR. The first two are useful for the mobility they provide, second two are for defense, and the last two are damage. Right off the top, I'm going to say Swift is terrible on Veigar. He just isn't one of the (Very few) champs that can make use of the perma +20 ms. Veigar will be getting in, bursting someone, then getting out. The only time Swiftness could have any advantage over Mobi is when you need to run, and I can guarantee that 20 MS won't save you. Skipping over Mobi for now, Tabi vs. Mercs. Tabi will never, ever save you. 20 armor and 10% less damage or whatever it is will never save you, they are useless to Veigar. Mercs, on the other hand, might let you escape from a nasty piece of cc that would kill you. However, if someone's in range to cc you, you're probably dead anyways. If you're going to get Mercs, get them during laning when the little MR they give is relevant. So your real choices are down to Mobi, Sorc, and CDR. Mobi are good if you're running everywhere and scoring kills up in top or bot. Otherwise, it's better to sit in mid and farm. Sorc vs. CDR is one of the biggest debates. The tipping factor in this mess is Void Staff. As we said above, it's pretty much core. Because of the way Magic Resist reductions are calculated (Flat Reduction > % Reduction > Flat Pen > % Pen) Sorc Shoes and Void Staff conflict, reducing each others effectiveness. CDR, on the other hand, is only ever reduced in strength by having 40% CDR already. So, we have resolved the conflict of CDR boots vs. MPen boots, and CDR wins. That's 3 items, half our inventory.
Next up, DFG. I really shouldn't have to justify this. 50% of their hp with the press of a button (I put it in slot 2 or 3, and then set it to smartcast. Smartcasting items is really handy.) is crazy good. Veigar gets free AP. DFG scales with AP. DFG is also what makes half a dozen "AP" champs work. Etc, etc, I've already had my rant on it. Basically, DFG is a good buy. 4 items.
And... We don't really have any other pressing needs. Most champs would just get more AP for damage, but Veigar doesn't need that. Instead, he likes to not die, so let's build for that. Let's look at the survival options APs have. Zhonya's (AP, Armor), Abyssal (AP, MR), Rylais (AP, HP, CC), RoA (AP, HP, Mana), and then other tank items, such as FoN, Warmogs, GA, Aegis, etc. Let's look at the AP options first. Zhonya's has a really sweet active, but it isn't going to save you from an angry 4 man mob. The AP and Armor, are nice, but not amazing. Abyssal, on the other hand, would be a good match, but it has the same issue as Sorc Shoes if you were to get into melee range somehow, and the MR isn't great, nor is the AP. Rylais and RoA, both give 80 AP, 500 or so HP, and a nice slow or large amount of mana respectively. The issue with RoA, is that now that a jungler is all but guaranteed, so is blue buff. Between bluebuff and your passive, you shouldn't have a need for more mana, and if you did, mana regen rather than full mana would suit Veigar better. So it's a slow versus mana that you don't have a use for. So, Rylai's is the clear winner of AP tank items. However, there are some *great* defensive options that don't offer any useful offensive stats for Veigar. Namely, GA and Warmogs. Warmogs, that godlike hp buff that it is, brings the stat that Veig needs most of all. 9000 armor and MR couldn't save him when he still only has 1500 HP or whatever it is. Warmogs can nearly double that. GA, on the other hand, gives a massive boost of defensive stats AND has that op as hell unique. So, which defensive items should you get? My personal preference is Rylais/Warmogs. Rylai's because it gives a nice amount of AP with 500 HP, and I chose Warmogs over GA because in unorganized games, you're not likely to see teams of 5 GAs, and as anybody who has played against bots can attest, 1 GA isn't a problem, just an extra 750 hp to melt. 5 GAs is a nightmare, though, because nobody stays dead ever.
For build order, take starting items > philo (If you started with one of the components, otherwise, skip it.) > Kages > Boots 1 (If you didn't start them) > Giant's Belt (Only If you're having trouble with dying in lane. Don't bother upgrading it just yet, you only get 70 HP and 80 AP) > Ionian Boots > Rabadon's > Void Staff > DFG > Rylai's > Sell Philo > Warmog's. At any point past completing Rabadon's, you should always have a blue potion running.
*Pauses for breath*
Oh god this is getting long so I'll take the cheaty way out and say Nax has it mostly right as far as playstyle goes.