I've been playing Veigar for a while now, and I really haven't seen the build that I use to dominate people. The trick is to keep sustain up. Veigar doesn't really need to build that much AP, and for this reason I have even won most of my ranked games as tank veigar (a build that no one will believe and I will not bother going into). Still, you want him to be tough AND powerful.

These runes are pretty typical for Veigar builds, I've noticed. Ability power per level glyphs, mana regen seals, magic pen marks, flat ability power quints. The seals and quints help with the early game, and the marks and glyphs will help mid-late game. I'm not going to stress this section because this is probably what you're already doing. Same with the masteries here, pretty straight-forward; not really anything you haven't seen before.

This item build is very important, especially the early-game chalice. Since you will be going mid, you are probably going to face an AP champ, making the magic resist irreplaceable. As an added bonus, you'll notice that your mana bar will always leave you with enough to cast a few spells, and you can feel free to harass because if your mana falls below a certain point, it replenishes very quickly. This is of course because of the passive on chalice of harmony stacking with Veigar's passive mana regen. Since you'll be bringing teleport, you can really go back after you hit 500 gp, but unless you're short on mana and hp, I would try to hit about 900 gp so that you can get tier 1 boots and at least 1 extra healing potion.

I do normally recommend sorcerer's shoes instead of Ionian boots of lucidity. Either one can be made viable, but I've found that having a little more burst on your harass is more effective than more frequent harasss. As a Veigar player, you really need to be an opportunist.

Rod of Ages will give your build some space if you're getting over-harassed. Oftentimes if I'm being harassed too hard mid, I will go back to base and get a ruby crystal with my tier 1 boots instead of rushing sorcerer's shoes right away. Finishing your catalyst will essentially fix any mana problems if you were still having any at this point. I usually obtain my RoA between 17-19 minutes, which still gives it some times to grow (unless one team is already steamrolling and it will be a 20 minute surrender). Rod of ages also gives me a unique kind of extra burst. The AP per minute on it along with my baleful strike stacks allows me to obtain AP fairly quickly without obtaining any items. People will wonder how I'm suddenly dealing so much extra damage in lane without having to recall.

Rabadon's deathcap is required for longer games. Since it adds damage from your baleful strike and you already have sustain, there's really no reason you shouldn't buy one. By the time you get one, your ap should be in the mid 500s, making it easy for you to nuke down enemies. Once I obtain a deathcap, I usually test my damage with harass, and if it looks like my baleful strike does a LOT of damage, I may try to one-shot their support or AP carry.

If you have that one feeder and the game really tapers on, you'll want to get a void staff. The magic penetration on it will allow you to hit harder on people that will be trying to counter-build you.

If the game just really lasts forever (or sometimes when I just get rediculously fed), I would buy lich bane. Not only will this give a very strong physical damage nuke with your combo, it will also allow you to DESTROY TOWERS. Teach your enemy that the game is going to end one way or another, and that it's only going to take you ~10 seconds to go through a tower. Remember when you're hitting towers with this to space your spells out, and sometimes it will be beneficial to use dark matter or event horizon with no target just to proc your lich bane.

I can't stress enough how much you really need teleport and ignite. I know that a LOT of you are used to using flash with Veigar, so you may end up doing better in the short run with that. However, I strongly urge you to bring teleport/ignite and learn how to use event horizon to stun enemies chasing you. Teleport gives veigar the time he needs to go back to base early game without losing harass, it will allow you switch lanes if they put their ap carry top, and it a veigar ganking by using teleport is almost impossible to beat

Just wait until some ruckus is going on and then use teleport when the enemy pushes past your minions at your allies. You'd be surprised how many times the enemy team will fail to notice teleport when a lot of spells are going on. However, this can be true about your friends as well. Your allies need to know when you're warping in. If it's easier, you can walk down bot lane to gank and then teleport back to mid. I usually only do this if my bot lane really needs help, I can't afford for mid to be pushed, and I don't care if bot enemies see me coming.

I put in this build that you should start with baleful strike with is normally true. However, if you are invading enemy jungle, you should put your first level into event horizon. Normally though, you want to put your first level into baleful strike and start getting stacks with it ASAP. Normally by end game I have roughly ~120-150 AP from baleful strike, depending on kills. Since this and sorcerer's shoes are going to be your only damage output for a while, be sure to get as many stacks as you can, even if it means you get harassed a little. Remember, you brought teleport, so if your enemy surprises you with something, you can go back, get your chalice, and come right back to lane with MR, mana regen, and a harshly vindictive attitude. After you build that chalice, the enemy harass will seem a lot less meaningful.

Once you hit about level 4 and get your first level in dark matter, I would recommend using your stun wall to lock them and then using dark matter and baleful strike to hit hard. If you think you can kill them doing that before level 6, use ignite to finish the job. Whenever it's up, I recommend using ignite when you are utilizing a large amount of burst damage, especially your ult. You may be surprised to find all the times you can stop someone's soraka ult or heal summonner spell from taking a meaningful effect.

Mid-game, you really need to get a pair... of items to build into a rod of ages. Don't be like other Veigars that are just waiting for their energy balls to drop. You have to get out there and stun them for dark matter to mean anything. Don't forget too that dark matter's end range will most likely out-range any mid champ you're fighting. Although it technically only has a range of 600, that is only to the center of the stun, and the outer boundary can reach over 1000 if placed correctly.

Toward the end of the game, you should be a tough but extremely powerful mage. Be an opportunist, use teleport defensively early game and offensively late game. Oh, and don't forget to set them on fire!