Ok this is a build I have used with Veigar. He is a late powerhouse. Although his power dosnt lie in his items so much as your early farming with Baleful Strike. Now if you have the ability to final hit meny of the minions with that spell. You will in the end walk away with an amazingly high AP. When I say high AP im meaning 900-1000 atleast... now the build its self is pretty basic and a little costly. However if you use Dark Matter on Hords of minions you will be able to farm a lot of guys and make quick cash.

These are all AP based for 1 soul reason... when your ap is that high just hide behind your allies and when they decide to take on well anyone just blast them and they will fall... rathery quickly if you would prefer to swap some of those out. Some other options for runes are
Greater Mark of Force, Greater Mark of Insight for Marks. Greater Seal of Force for a Seal change. Greater Glyph of Insight, Greater Glyph of Force for a swap out in glyph's. And either Greater Quintessence of Insight, or Greater Quintessence of Force to change out quintessence's. These changes will increase you Magic Penetration OR Increase your AP at level 18 substancialy almost double what the standard ones would although it probly wont make that much of a change at level 18.

Pros

High damage out put with his spells
Great for support
Amazing lane control


Cons

Squishy
Slow start
Costly

These items basically speak for them selves. You start the build with a way to gain mana. Using Baleful Striketo boost your AP. After you get your boots you dive stright into building your AP the only reason I have put 2 Rabadon's in there is because that 30% boost mixed with the 155 AP boost almost equals out to around 500 AP in 2 items Plus the ap you have managed to get With Baleful Strike.

Well this is pretty stright forward
1 kill as meny minions as u possably can with Baleful Strike
2 once level 8 or higher with a relitivly high AP use Dark Matter to kill large groups which will earn you a very wealthy sum of money.

Ok for the Summoner Spells I prefer to choose Teleport and Ghost.

Ghost for the ability to catch and kill any person running with your high AP you should be able to either 1 stun them long enough to finish them off or 2 just cast the final spell needed to end their life. (This could be swapped out for Flash if you find you work better with Flash.)

Teleport gives you the option to go ANYWHERE on the map gives you way more freedom of movment and just helps you all round

As for all the other spells I personaly havent found any that help you for instance Smite wont do a thing for you since you SHOULDN'T be Jungling with this guy. As for all others again I just havnt found them to be of any use you hit so hard already that even if someone is trying to go toe to toe with you. They will lose health much faster unless you get stunned.

well the skill's are pretty simple

Baleful Strike
Last hit every minion you can find to build up your AP fast. The more last hits you get with it the more AP because every kill you get +1 AP and if you can manage to take down a few Champs you get anywhere from 1-5 AP depending on the levle of the skill.

Dark Matter
Again a simple spell to use basicly use this for lane control and hit champs with it as well as minions it can deal a great amount of damage. Although you will need to predict where the chapion is going.

Event Horizon
This one is great it will stun any champion that walks through the walls so the best option is spar a little with the campion get them to about Half HP and use this stun them in place. Blast them with first Dark Matter, Then Baleful Strike, and if you have Primordial Burst at the end use it and 9/10 times they will fall. This spell can also be used as a great get away tactic drop it behind you as your running away the person will be stunned and you can make a clean get away.

Primordial Burst
This one is pretty self explanitory use this on every champion you can find because first you deal standard damage + your AP in damage plus 80% of the champion your attacking in AP damage. So for instance you shoot this a Malphite well it wont do a large amount of damage because Malphite is normaly built as a take not an AP user. However take Annie or Sivir or a character of the like that thrives on AP and hit of that could deal well over 2000 damage in the end.

Ok so if you dont feel like reading ALL the other parts which are pretty small then ill break it down here. First and for most BUILD YOUR AP I cant stress that enough Veigar dose burst damage that can melt a champion. Secondly your mana will be low to begin with but after a little while you will have an substantial amount and with his passive. The ability to regen mana at .75 faster per 1% missing late game mana will mean nothing to you. Now you may wana swap out the boots in this build for Ionian Boots of Lucidity. From there just act as a support to a team mate tank just blast the heck outa the enemy and all should end well for you.