Hello, Veigar is my favorite character for a variety of reasons. He makes people go boom and people dont even realize what happened to them. If played correctly you will be a nightmare for the other team and a great support with your stun.

Pros
Great farmer
Anti-AP
Very high damage nukes
Great support with stun

Cons
Squishy
Weak against high MR tanks.

First you want to start with Mekai's Pendant and two health potions. My first trip back I buy Tear's of Goddess and Boots of Speed. After boots I go for Megai's as this is when you start killing people. If your team is not doing well then skip Megai's. If you are farming/killing well after Megai's go straight for Rabbadon's if not just finish Archangel's. You can either go Voidstaff or Deathfire's Grasp at this point. I prefer Deathfire's as it is an on demand 1 minute extra nuke for you. Now you finish off your build with a Banshee's Veil and potions.

For marks I get magic pen for obvious reasons.

For seals I go -cds you can go for health or mp5 but really if you play Veigar correctly you wont need them.

For glyphs and quintessences I just go straight ap.

You want to max out Baleful Strike to make last hitting minions easier. Start out with 2 in Event Horizon as it secures an almost 100% chance dark matter lands with the exception of people with Mercury treads. I get dark matter at lvl 8 and max it after maxing baleful.

Your Skill order for any situation should be E-W-Q-R or E-W-Q. All you should be doing is stunning and doing W-Q. You should never be in the front and always be in and out with your stun and nuke sequence. You should always be targetting squishies with your nukes.

Flash and Ghost is the best Summoner Spells hands down. Teleport is a good alternate but in my opinion Flash and Ghost is the best because your goal as Veigar is to survive and nuke.

You should always try to solo because you aren't going to help much with damage in the beginning and is hard to farm with another lanining partner. When solo don't worry about harassing too much because you won't be doing much damage and just wasting your mana on baleful last hits. You should always stay back and come in to last hit a creep and always try to last hit with baleful. If they are harassing you too much just ask for a gank from your jungle.

When you have Tears of the Goddess and Megai's you should be able to harass and land some kills now. The way I kill people is i E-W-Q run away rinse and repeat until they are in range for me to kill them with E-W-Q-R. To help your farm you should dark matter the casters in the back and at about lvl 12 you should be able to one shot them with dark matter. When you get around 300 AP you should dark matter the jungle creeps from the mid lane.

Remember that Veigar is a very good farm hero and without that farm you aren't going to be able to kill much. Always play very passive in the beginning and try to use all your mana to kill with baleful strike.

You are there for your team to stun and nuke. You never lead your team unless its for a gank which you should be stunning first. When you are confident that you have farmed enough go around the whole map and go gank. Always ask for blue buff because that buff really makes Veigar shine. When you engage in any fight all you should be doing is stunning then nuke then run till your stun is back up. If played correctly no hero should ever touch you while your team is tanking for you. People won't even know that they died because Veigar can kill someone that fast.

Veigar overall is one of the strongest heroes and is extremely fun when u have enough AP to 1 nuke someone. Just play very cautiously and you will be blowing people up in no time.