I've been playing LoL for a few months and Veigar is my go to. Played DoTA before LoL too.
I play Veigar as the "come near me, and you'll die" character. It is commonplace for me to nuke a tank in less than a second late game. The advantages to this is that the notion that I can focus any of the enemy team and kill them quick keeps them all at bay, and allows my team to play with them on our terms. The bad, is that I'm dependent on my spells not being on cool down or I'm toast - hence why my build gives you 32% CDR (leaving room for blue to max it).
Veigar is a glass cannon, play him like one.
(He runs around flailing his arms for goodness' sake).
Keep your distance; never get hit be creeps unnecessarily (unless it is to help you max your Q farm); harass; farm your AP with Baleful strike, learn to pop all 5 spells (counting Death Fire Grasp [Herein DFG]) in rapid succession and ALWAYS keep your distance.
I repeat keep your distance because that is one of the better defenses. I've killed champs because they have stupidly moved close to the creeps to last hit one and got too close to me. I have stayed in team fights with less than 200 hit points because I am able to keep my distance and only close for kills.
Don't take unnecessary damage - basic premise of the game, less important on tanks, more important on little Veigar.
Harrass - learn to do it by stunning, casting meteor and leaving meteor timing is such that you should be able to stun; cast a meteor and they will be hit before they can move again. Do this once or twice on a champ and they are ready to be killed. You can also hit them with a baleful strike after casting meteor, just don't gimp your AP farm.
Baleful strike - do NOT use this as your general harass (that is your meteor) or you risk gimping yourself late game. Early game I will cast this until I am out of mana, and then (thanks to his passive) continue to cast it as often as I can to maximize his AP. I have pulled 60 AP before my first back. You won't need to back early if you farm this because it will adjust your damage accordingly.
:IMPORTANT NOTE: Baleful strike is WEAK at level 1, make sure that when you hit the creep with it the creep will die. A good strategy to ensure this when fighting with your creeps for last hits, is to keep moving Veigar until a enemy creep is low health, then shoot a basic attack and immediately follow it with a baleful strike.
Rapid spell casting - I can't stress the importance of this enough. I have killed assassin class champs (without popping stun to stop them) that were chasing me because I was able to cast DFG, baleful strike and my ultimate all within milliseconds of each other.
I don't roam the jungle by myself. I try to carry lots of wards and bush check with them late game. I typically only blue with someone so I don't get caught with my pants down because everything is on CD.
I will often run around with a tank, most of the game to provide the extra safety that is necessary when playing such a squishy glass cannon.
In those seconds before team fight, try to lock down members of the enemy team in close proximity to each other and simply drop a meteor on them. This weakens their team both in hit points and in resolve.
Another great tactic with team work is if any members of your team have a Will of the Ancients, run with them - with the high damage output, you will heal your self with very few spells.
I have only offensive items - as you might have noticed I hold the view that Veigar should be able to kill people easy. I don't build any defensive items as most of the guides do on here, as truly the best defense is a good offense. I have killed many Warwicks right after the ultimate on me but before they can finish off my meager few remaining hit points in less than a half second because of solid execution of powerful spells.
Don't build defensive items, Veigar doesn't have the hit point pool to make defense and magic resist useful, and usually if you die, being resurrected in 3 seconds won't help. The times I die as Veigar is typically because 3 or more members of the enemy team jump me all at once.
I mentioned elsewhere about wards - Extremely important. Knowing where your enemies are is the difference between you killing them and vice versa. I played one match against a Twitch and an Evelyn that I had decided to sacrifice an Archangels simply so I could carry around vision wards (yes I could've gotten an oracles, which is better in some cases, but not all). There was an instance that both of these stealthy champs came along to jump me, however I had a vision ward in place, which allowed me to see them, stun them, drop a meteor on both, ultimate the one with more HP and Baleful strike the one with less HP and walk away having taken no damage and gotten a double kill. Moral of the story - wards save your bacon.
I have gotten Veigar up to 1250 AP, with a Q farm of roughly 175. Don't ignore your Q farm, particularly early game. I have had games where I didn't back for the first time until I had nearly 3k gold because I had a high Q farm and was still killing enemy champs.
One more reason I focus AP with Veigar, is that meteor in team fights is a ridiculous ability for two reasons. One, the enemy team often misses the ground effect warning and two, it can make his ultimate an execute for their AP carry.
Veigar's stun... Learn to use it, and well. don't pull the nooby mistake of casting it centered on someone. Learn to land the edge on someone at full range. I have laned against good ad carrys on mid, and because I could use the stun effectively I could wear them down and then pop clarity to kill them.
Ultimate - base damage + 1.2*AP + .8*(target AP) = dead champs. Don't use it unless you have to. I have wasted many ultimates because my Q killed the target and I cast them in tandem.
DFG - It is more like an ability than an item. It does 30% targets current health (+3.5% per 100 AP) as magic damage. This is part of the reason I build three archangels instead of two and a void staff. With 1200 AP this item alone will do 72% of targets current health as magic damage. Hit with this first.
1: is a no brainer - you need to start the Q farm. The only time I deviate from this is if we are trying to jump their jungler or some other member of their team, then I will grab stun first as a 2 second stun at level one is rather strong.
2: Event horizon if you got Baleful strike for the CC, and baleful strike if you got event horizon.
Beyond these - max Baleful strike ASAP and farm it. Every point of AP pulled from that ability costs you NO GOLD, to not optimize that is to waste your time.
Ultimate when it comes up, like all champs.
I will level event horizon a few times before leveling meteor to level two - after two meteor can wait for end game (by the point that I level meteor to level two, it will kill ranged creeps in one hit). While meteor has a lot of damage output, and is a great harass, it is unnecessary for kills and team fights. The stun is more useful, and on 1v1 fights, Death Fire Grasp, baleful strike and ultimate will kill most.
Cool down is extremely important. When I was first learning Veigar I toyed with the idea of selling DFG and buying a fourth Archangels, it gave me roughly an additional 200AP, but at the cost of having to wait much longer to cast. It was a lesson in diminishing returns.
Due to that experience I decided to build heavy cool down reduction with my runes. I go flat rate with seals because it provides an early game bonus for cool down. CDR per level for glyphs is best to balance your early game and late game effectiveness. Marks should be Spell Penetration (this one should be obvious). Lastly, Quints should be more flat rate CDR to aid you early game.
Pretty straight forward, save the 9 points to utility. Move speed is important to keeping this little dude alive, hence why I put four points into that. I put one point a point into the runic affinity because having 20% more time with blue is far more useful than say, 1% spell vamp.
Clarity and Flash.
Clarity gives Veigar the ability to stay in lane longer, and provide more damage. It also allows him to farm AP with Baleful strike longer early game. I've also used it to lure in enemy champs that think I'm out of mana, only to use Clarity and kill them. Clarity is more important and useful to Veigar than ANY other summoner. Don't underestimate it. Late game it adds the benefit of staying in long fights long after someone without Clarity would've ran out of mana.
Flash can be used both offensively and defensively. Offensively, it is extremely powerful tethered with Veigar's stun and damage output. I have killed champs who were trying to keep a far distance from me in lane by flashing towards them and unleashing the hellfire.
Any other summoner spell on Veigar is pointless.
I often see guides giving Veigar ignite - I ask WHY? If you need help for the last 100 damage, you shouldn't have built a Zhonya's Hourglass.
I've seen Teleport in other guides - which is great, but it's worth hoofing it everywhere for the advantages Clarity and Flash give.