Veigar I would say, is the highest nuke in the game. The way I build veigar is very passive early game, but very aggressive late game. He may not be a carry in his attributes, but thats how I play him.

Please read the entire guide to get the full extent of how I play veigar. Building him like this works, but no where near as well if you don't follow my gameplay.

I started with AP blues and Quints, but after I got totally destroyed by another veigar I asked him what he had. He said cooldown blues and quints, so he could farm his Q ability early game. All of the others are standard, mana regen because it makes a difference. Magic pen because I'm an AP guy who wants to nuke everything.

The Q is there for a reason. If you read the ability you gain 1 AP for every unit this spell kills, and 5 AP for every champion. THIS IS EVERYTHING. This is what makes veigar so much more of a carry late game, and also what makes him hit so hard without and AP items. If you use this Q the entire game, for all your minions kills, you should have 180+ AP. Of course I don't expect this of you, I don't think anyone takes that much time to use his Q.

On to more important things, you want to level up your Q to 4 as fast as you can. Get 1 point in the stun, then 1 point in the dark matter, but continue to get the Q. After it's level 4, killing minions with your Q to level up your AP will be quite easy, but it still needs your attention. Don't just dark matter the minion wave, take the time to last hit each one with your Q as much as possible. People do not understand how effective this is, mostly because they don't take the time to actually do it.

After your Q is to level 4, get your stun at least to 4, it increases the time it stuns every time you level it, and you want the longest stun possible. After that work on your dark matter again so you can properly nuke. The stun does not last long enough at level one for people to get hit by your dark matter, so only use it if they're on your tower. Your combo will get anyone squish from half their health, to dead or really close to it after level 14.

This is what makes Veigar so intense. This is why I have 1250 AP with a full build. His early game.

Veigar takes mid. No questions asked. End of story. Farming your Q to get you more permanent AP is priority #1. Take your mid, and play PASSIVE. Yes, I know it's a term you don't see often in any of these guides, but it's something that makes the world of a difference. If you use an ability, and it's not your Q, you better have a damn good reason for it. Save every last bit of your mana, and mana pots all for your Q. Yes, the enemy may be shooting your tower, but you keep farming your Q, don't wipe out the wave with your dark matter, or try to harass with your stun combo. Farm. Last hit. IMPORTANCE MAXIMUM.

Many people who go mid, play mid often and see veigars that farm their Q properly. They know if they stand between you and your minions you will lose in the end. This is the only time I will permit you to use your combos. Hit them, and hit them hard if they stand in your way. It's better to make them scared of you and lose only a little AP in the end, then to sit back at your turret like a dead cat. You need to farm the Q, and if they get in the way, make them pay.

While mid, you go through a lot of Q's, especially since you have the cd runes to help you early game. If you run out of pots and mana, port back and teleport back in. By this time you should have enough gold to buy a Tear of the Goddess. After you have that, your mana explodes and you barely never run out, and always have enough for a stun at least.

Veigar is not an early game character. He doesn't pop in, nuke, kill, and pop out like LeBlanc, so don't expect to be doing that. Stand strong, but don't go for any kills because your cooldowns are long enough for them to escape and so you can't escape. Only go in if you have a second player for the gank. Sometimes people do get used to your passiveness and stay middle even with a third of their health left. Go kill them, if you can hit them hard enough to kill them with one combo you're golden, but otherwise just wait till mid to late game for ownage.

Slow Game:
Continue to farm your Q. Find a minion wave, and follow it back to your tower farming your Q as much as possible. This is important when there's nothing going on. If they want to play passive, power yourself up. Don't waste time.

Teamfights:
Find your opponent. This is key. "Veigar can nuke any AP person, it's his job!" I hear this too often. With veigar, your target is NOT always someone with AP. That should only be an extra perk, "Oh he's got 150 AP, I guess I'm just going to hit him that much harder!" Thats all his ultimate is, none of this, "He's not AP, I can't do anything!" When an ashe runs up to you and starts kiting you across the map, don't run! turn around and nuke her! If an ashe has no defensive items you should be able to combo her from full health to none instantly. No questions asked. With a full build, Veigar's ultimate should do 2000 damage + 80% of the targets AP. Yea, 2000 damage on its own, not including the opponents AP. Thats a lot, more than ashe's health without proper items. Teamfights are Veigar's happy time. This is when veigar can roam free and not have to worry about anyone killing him. You just need to wait for the teamfight, don't initiate. If you can, make sure you can get out of it as fast as you can so no one is on top of you. Veigar likes his enemies at a safe, controllable distance. His cooldowns by now are great enough so that he can stay in the fight for the entire fight, but make sure you are on the outside, you have range.

Being focused:
Stay away! let all of your other teammates go in there first and then go in when you know you can stun them before they realize you've joined. If they want you, don't go back in there! Wait for your stun, practicing with this is the hardest thing to do, but one of the most important things. Your stun is amazing, because if affects a lot of people if placed right, and its ranged. Learn how to stun people with the edges right when you put it down, don't surround them with it because it gives them time to dodge, time to hurt you, time to flash out, or use their own abilities to blink out. It's hard to get used to, just keep trying.