Veigar is a really powerful champion if you utilise his abilities correctly. His skill set works fine for bursting down one enemy (1v1), but what many don't understands is that Veigar is extremely powerfull in team fights where he can ingage with a massive stun (that last for 2.5 seconds) that can stun the hole enemy team. Veigar can use the same stun to disengage or save himself or teammates from tricky situations.

Veigars ultimate even scales with the enemies ability power! This means that Veigar can burst down any enemy AP champion that gets fed, witch allows you to make big comebacks. If the enemy team has more then one AP champion Veigar will always be viable.

As Veigar its most unlikley that you will be able to kill an enemy in the early laning phase (not impossible though). Veigar is a stacking monster, so the 20-30 ability power you can get from runes is negligible when Veigar easily get ~ 500 AP mid game, and ~ 1000 AP late game. The extra magic penetration you get from the runes will increase you damage sooo much against enemies squishies, so worth.

Pretty standard masteries for AP champ, you can adjust the masteries after your liking. Maybe give Veigar some more tankyness if you feel like it.

Get Doran's ring and health potions for some extra sustain in lane. Veigar do have problems with mana in the early game so try to be careful with your abilities.

Get the chalice at the first back so you get more mana regeneration so that you can spam your abilities a lot more and poke you enemy in lane. The boots are self-explanatory (Speed = good).

Finish Athene's to get some ability power, mana regeneration and magic resist. When you have this item you should be able to stun your enemy whenever you have the opportunity and give them a good poke with your q and w.

Rabadons synergise well with your passive from your q. Void staff gives you magic penetration that increases your overall damage, and so will the boots as well.

Get mobility boots instead of the magic penetration boots if you want to catch your enemies and stun them with you e.

Max Baleful strike (Q) first so you can stack ability power. This ability is great for last hitting and harassing.

Then I like to max Event Horizon because (E) it reduces it's cooldown and increases the stun duration witch is super great for team fights .

I max Dark Matter (W) last because it has a high AP ratio (100%) and scales well with just pure items and Veigars Q passive.

Put a point in Primordial Burst whenever you can, this is the ability that will set fear in your enemies.

Event Horizon (E) is the ability that makes Veigar so powerful, learn to use this ability and you will success.

In the lane you should stun your enemy whenever you want to trade, when he is stunned you can use your combo: W, Q, auto attack . When the enemy is low enough you can go all in with your ignite.

If you suspect a gank is on the way, save your E so you can escape.

You can use your Q to poke the enemy.
Don't use your W on moving champions, since it has a delay and is easily dodged. You should always stun the enemy with E then you can use your W.