DISCLAIMER: This guide was made in January of 2010. Items, champions, summoner spells, and the metagame have changed since then. Take this fact into account.
BS: Baleful Strike
DM: Dark Matter
EH: Event Horizon
PB: Primordial Burst
Veigar tends to be overlooked because of casters like Anivia, Karthus, and Fiddlesticks. Honestly, Veigar can be the most powerful caster due to his scaling AP because of Baleful Strike and Primordial Burst. When played right, two- or three-hitting enemies is quite common. When played wrong, he's a squishy free kill.
Veigar's main skill. Throughout the game, this should be used to farm AP on minions constantly. Due to its short cooldown, it should also be used to remove any spell shields.
Powerful, but useless, except for farming creeps, without Event Horizon being a decent level. Try to cast this BEFORE Event Horizon to make sure it hits.
Veigar's only CC, which makes it very important. If you want to be good at Veigar, you must learn how to vitalize Horizon's strange hit-box.
Massive damage champion finisher. Because of its massive mana cost and great rewards for kill shots, avoid wasting this as a non-killshot at all costs.
Veigar is different from all others in the fact that he can increase his AP without items or masteries. You must utilize this ability constantly, using Baleful Strike to farm minions and Primordial Burst to last-hit champions. Also, this build is not for soloing mid.
Farm farm farm. Use BS to last hit minions any time you can. Ignore your lane-mate if they rage that you're wasting all your mana. Help your lane-mate try to kill enemies if you must, but that's not your incentive right now. Auto-attack melee champs to get them to back off if need be, but that should be the extent of your griefing. Remember to play conservatively and dodge anything coming your way.
By now, your Mejai's should be up and running so you should be ganking, right? Wrong. Keep farming, but now you can afford griefing enemy champions with your Baleful Strike because of your Tear of the Goddess.
Level 2 boots, Rod of Ages, Tear of the Goddess, and Mejai's. It's fun time baby. Start putting the smack down across the map, also being sure to grab golem any time you can. And also, don't suicide or make stupid mistakes. Your Mejai's stacks are valuable.
If you're doing bad, you should be mostly just a small nuisance by now. If you're doing good, you should be the terror of the game. Destroy everyone in your path while you build up your high AP items that are still missing. Once these items come into play, you'll be Veigar: the Tiny Master of 700-1000 AP (aka Death incarnate). Remember: no suiciding or stupid mistakes.
Though overlooked (and hated because of his voice), Veigar is a menacing threat of mass destruction when done correctly.