This build is designed for a fully offensive Veigar; you're prioritising getting an insane AP spike rather than trying to builk up. The Rod Of Ages gives you this gradual spike, hence why I have emphasised it as your core item.

I've tried to make this guide tailored to my personal taste - basically, I've tried to avoid advice such as "just have your jungler gank more often" because I know you can't always do things by-and-by.

The main things to keep in mind are as follows;

1. Veigar is exceptionally Mana-hungry in the early game. Always demand (ask politely for) the second blue buff.
2. Veigar's power grows exponentially if you farm with his Q correctly. Even enemies stacking Magic Resist won't be able to handle the sheer volume of magic damage without having to sacrifice armour (which means your ADC will have a breeze of a time). Just to be clear; by 40 minutes, you can have over 1000 AP.
3. Veigar is a mage killer. He is one of the best posssible counters for any enemy that builds AP. If you're against a mage in lane, focus them in fights; you can take them 100-0 with a single E-W-Q-R-Q combo.

Take Q, then W, then E. Max R at 6, 11, and 16, when it's available. Max Q first, then W, then E.
At level 8, you have to take either W or E, cause you can't take Q. I take W for the wave-clear potential, but take the E if you need a marginally longer stun.

Equilibrium - This ability stacks with other abilities that restore mana, such as Mana Potion or Chalice of Harmony. It encourages you to get a large pool of mana, and it helps you out if there's a minute or two of downtime between fights.

Baleful Strike - The absolute core of what makes Veigar so dangerous in the mid-late game. When your ultimate is down, this is what you'll be using to finish off enemy champions, due to its natural low cooldown, and the bonus from killing champions.
Early game, use Baleful Strike to farm one or two minions per wave; trying to poke too much with it, or using it too much in general, will waste your mana.
Any chance you get, use it to last-hit minions in the mid/late game. At many stages in the game, Baleful Strike will be able to one-shot normal minions. Focus on last-hitting big minions as well.
If you're good at taking advantage of the passive, and get all the items you need, expect to have, overall, around 600-800 AP in the late game. If you're doing fantastically, you'll have over 1000 AP.
Why you should try to last hit champions with this; you gain 2 AP from the active, plus 1-5 AP from the passive. A jump of 7 AP per kill makes a huge difference.

Dark Matter - The Nuke spell; timing is key; place it where a minion wave is going to be for maximum damage spread. At several points in the game, this can kill an entire minion wave (sans the big minion). If an enemy activates a Zhonya's, or is otherwise invulnerable/untargetable but going to be in a predictable spot, throw this down right before their effect ends.

Event Horizon - Your huge crowd control device, this skill has 3 functions, all of which it can do effectively at its 1st level: It can initiate fights, by trapping enemies after they get too close, it can split up an enemy team during a team fight, by plonking it down right in the middle, and it can also be used to escape fights (by timing it right, you'll insta-trap the enemy as they walk towards you, and you can hop away merrily). The basic combo is to place your E, spam your W on the enemy's head, and if you can, throw in a Q - that order is important, as the W has a lag-time before it hits, whereas the Q travels straight to the enemy.

Primordial Burst - The mage killer. This is why enemy mages fear Veigar. It does 120% of your OWN ability power as bonus damage. plus 80% of the target's ability power. This means that, especially early on, you should always focus on the enemy mage. Personally, I'm very cautious about throwing this one around, and only use it if the enemy mage has around ~50% hp or less, to ensure either it kills them outright, or that I can end them with a Q immediately after. Primordial Burst travels in the same manner as his Q, meaning it can be blocked by a Zhonya's, or the target otherwise becoming untargetable; for this reason, in the late game I wait as long as I possibly can to use it, baiting out a Zhonha's using the Q.