Skills and Order of Leveling
"Veigar's mana regen is increased by 0.75% for each 1% of mana missing."
This passive is great for staying in lane and spamming your abilities often to farm Baleful Strike (Q). Veigar still has some mana issues early in the game, so you will probably need to build at least one mana item (Tear of the Goddess into Archangel's Staff). More on the build in a minute.
Baleful Strike (Q)
"Unleashes dark energy at target enemy, dealing magic damage. If a unit is killed, Veigar gains 1 ability power (1-5 per champion kill)."
By farming your Q, at the end of the game you can easily get 500+ AP and 750+ for even longer games. If you land all your spells on a single or group of targets, you will be able to easily burst down multiple champions from full health. The key is to getting good farm with Q in lane so that late game, you will have enough AP to deal tons of damage.
Dark Matter (W)
"Veigar calls a great mass of dark matter to fall from the sky to the target location. After 1.5 seconds, it deals 120-320 (+1.0 AP) magic damage when it lands."
Without landing your stun (E) or another form of CC, Dark Matter is quite difficult to land as it is very easy to avoid it's small, but deadly AOE. By late game, enemies caght in your W will be sorry as it has ridiculous scaling (1.0 times your AP). It is great for clearing minion waves and just dealing enormous AOE damage in general.
Event Horizon (E)
"Veigar twists the edges of space around the target location for 3 seconds, stunning enemies who pass through the perimeter for 1.5/1.75/2/2.25/2.5 seconds."
Veigar's AOE stun is a large AOE skill shot that requires good placement and movement prediction on your part. You must anticipate the direction enemies caught in the ring will run so that you know where to cast Dark Matter (W). Event Horizon is great for initiating on enemies that have strayed a little too far from the rest of the group. It has a pretty long cooldown (16-20 seconds), so if you miss, tell your team to backoff and try again.
Primordial Burst (R)
"Blasts an enemy champion, dealing 250/375/500 (+1.2) plus 80% of his target's AP in magic damage."
Veigar's ultimate is what makes him the anti-carry carry for your team. Not only does it scale at 1.2 times your own AP, but also 0.8 times the enemy's AP as well. Add that to the base 500 magic damage and any AP target (or non-AP) will be decimated! With this one ability and your Q, you should be able to nuke down an enemy at full HP.
For the first four levels, level Q, E, Q, and then W. After that, prioritize Q, then W, then E, of course taking your Ultimate at levels 6, 11, and 16.
This skilling order for Vegiar will allow you to have all of his primary abilities by level 4 and then allow you to farm your Q as fast as possible so that late game you will have plenty of AP.