Changed Shurelya's Reverie to Eleisa's Miracle since there was too much Cooldown Reduction that brought it over the cap.

I wanted to have Veigar do a lot of damage quickly. To do that you he needs AP and cooldown reduction. While he remains squishy, he can stay behind the minions and hit the enemy with Dark Matter while he farms. Don't rush him unless you know you have the kill or he will die quickly.

With the runes I wanted to get his abilities to punch through while giving him some early game protection. The seals will also give him an early game advantage.

9 in offense to get that 15% magic penetration.
21 in Utility to maximize his Magic and cool downs.

I start with the Meki Pendant as it's cheap and gives you the ability to grab a couple potions right off the bat.
Next Boots of Speed to get you moving and a Philosopher's Stone to get some gold flow. After that grab the Sorcerer's Shoes then it's time to start working on your core.
Hit up Morello's Evil Tome followed by Will of the Ancients. This will be a good kick off to do some real damage. Always good to have a Rabadon's Deathcap Then convert that Philosopher's Stone to a Eleisa's Miracle for some regen and tenacity.
Round it out with a Deathfire's Grasp and you'll get the bonus damage too.

If you find things going well, I'd keep on with the Rabadon's Deathcaps as they have the most AP which is perfect for Veigar. If you find yourself too squishy, I'd suggest going with a Force of Nature for the extra magic resist and health regen. You'll sacrifice some AP, but it's better than being dead.

You'll want to get one of each early on, but really focus on Baleful Strike as that will be your primary ability choice too boost your AP with last hits. after that is maxed, I'd run a balance of Dark Mater and Event Horizon.

Veigar tends to be mana hungry especially early game. Having clarity will really help that out. Heal will keep him in game longer and have a little more survivability.

You could also do Teleport or Exhaust, thought they may not be the best choices.

this is my first build here, but not my first that I've done personally. let me know what you think.