I've gone and written a very extensive build, but my reasoning in it was off or vague almost everywhere, so I'll drop the gist of it concisely instead.
21/0/9 (ms trait from utility 3/3) 9 Magicpen, 9 Armor, 9 MR/level, 3 MS quints Teleport/Flash Doran's/Boots/Cloth - health pots (possibly mana pot) Athene's-Sorcerer's-Rabadon's-Void Staff-Warmogs/Righteous/Randuin's-Banshee's/Warmog's/Locket otIS
[ALWAYS attempt to lasthit 2 minions at once with Q, always try to lasthit with Q. 1AP is worth 21g. 1 Q lasthit is worth 51.] Abuse your strong level 1 by poking your enemy with Q while lasthitting, but get as much farm as possible (most likely you'll need it soon) Continue this in levels 2,3. Consider level 3 or 4 the time to hang back, get dived by enemy and jungler, barely survive, B, and TP back. Any farm you get in this period gives you more chance of victory and is a mistake from your enemy. If not pressured : B when you can get a Codex and boots/Chalice/Blasting Wand, TP back in, wreck. Midgame : use your mana regen to E-pressure, a halfdead champion means they lose if they all in you without being a Diana, Riven, etc. Roam bot, or top if AP (or dumb) top.
Lategame : (rush a giant's belt if picked often) spam E in engages, play the obvious way (QW anything that moves, deal as much damage as possible, R on either fleeing near-dead targets before they get away and heal, or key targets (Personal counters>Carry APC>ADC>Control/burst APC>CC festivals (Leona, Kennen, Malphite, etc.))
The added armor/MR/health and MS will allow you to float through teamfights rather than being scared on the edge of it. Generally, get enough health and resistance to counter the champion most likely to jump on you (Leblanc, for example : get >3K HP). Never forget to stack AP, it's free and worth so much. Never put yourself in picking situations without need. Teams can, and will, jump 3v1 on a splitpushing Veigar. If your teamwide turret siege is weak, and you are able to get close enough to the turret without dying right away, replace an item of choice (not rabadon, and keep up your CDR) with Lich Bane.
Most importantly, Veigar is wholly and utterly shut down by champions he can't blast (Ziggs, Xerath), by champions he can't predict and/or outplay (Talon, Annie, Leblanc), and by superior execution of champions that normally counter him but not necessarily the bruiser variant (Diana, Rengar). If remotely possible, don't, EVER, pick Veigar vs Talon or Leblanc. Besides those, anything can potentially be overcome, but these two will ensure you will not reach carry potential, will not use it, and will not be up 90% of the time lategame.