Not Updated For Current Season
Q[Baleful Strike]:Your first source of damage, always try to farm ap by lasthitting minion with this skill. When it's at lvl 5 it does a good damage to enemy and in team fight you should spam as you can. Always try to lasthit an enemy that is dieing.
W[Dark Matter]: It does an insane damage in teamfight, and it always hit combined with the stun.
E[Event Horizon]: The most useful skill of veigar, he stun all the champion passing through the barrier. Usefull in teamfight and good combo with many other's champion skill (like the ulti of nun, katarina and fiddle) and many times will save you and your team mate from death.
R[Promordial Burst]:A good lasthitter skill. It does a good damage (more against magic champion). You should use this to kill champion that have low life.
Teleport: it's a very useful skill. With this ability you can help to defend a tower, go pushing quickly and waste less possible time.
Flash: Very useful for veigar. You can use this to reach with your stun an escaping champion and combined with the stun will save you from death many times.
With this masteries you've some extra mana/mana regen., 9% cooldown reduction that is very nice and 15% magic penetration.
Red and quintessence: magic penetration (more damage from skill)
Yellow and blue: mana regen per level (with this runes combined with the passive skill you'll be almost never out of mana.
Item (buy order)
-Sorcerer's Boots: magic penetretion (more damage)
-Zhonya's Ring: Rush this item, it can save you in some situations and it will give you a lot of extra ap
-Deathfire Grasp: 15% cooldown and the passive skills of the item with the ap that veigar can reach will hit enemy very hardly.
-Lich Bane: More movement and the passive use the ap which veigar almost always has an high amount.
-Rylai's Crystal Scepter: More survivability, many ap, and the passive isn't useless
-Void Staff: more magic penetration
Early game: you do very little damage, so farm ap lasthitting minion, and if in the lane you have a lowlife champion you can try using your stun to kill him (only if your lane mate does a good damage).
Mid Game: Keep farming ap. You can try to kill enemy spamming on him q, and then when he has low life, stun + meteor, q and then r. (most of the enemy with low life will die with this combo.
Late Game: This is the part of the match in which veigar rules. Follow your team mate, don't go alone. In team fight try to hit many enemy as possible with the meteor, use your stun with intelligence and kill the enemies using your skill. If one is escaping you can use your flash and then the stun to chase him. In this part of the game to start damaging an enemy use the passive of deathfire that in this part of the game will hit for almost half life.
-If you're going to die remember to use zhonya!!!!
-Lasthit with q as possbile
-Steal the kill for earn ap !!!
-1 vs 1 skill sequence: deathfire, stun, meteor, baleful, and ulti