Veigar is a high burst champion that will has the ability to farm early game, and then nuke in late game. To offset his incredibly high damage, he is on of the squishiest champs in the game. By playing Veigar you will be a very pivotal champ in teamfights and almost always be the deciding factor.

For my marks, I get magic pen not to just do more damage to the tanks, but to weaken a mr in a dps champion that they got in either a Banshee's Veil or Mercury Treads.
For seals, I get flat mana per sec because veigar is very dependent on mana early game. This should solve that problem and give you enough mana to farm and harass a little bit
For glyphs, I chose cooldown reduction runes. Veigar's abilities are very powerful, and late game, baleful strike and dark matter will do insane amounts of damage. With cooldown reduction, Baleful Strike will have a measly cooldown of about 2 seconds and dark matter will have a cd of about four.
For quintessences, I also get flat ap runes because as I stated earlier, it will help farming with baleful strike a lot

My Masteries are 9-0-21, which is pretty much standard for a strong caster.

For attack, you need the 15% magic pen so upgrade 3 points in Archmage's Savvy and one in critical followed by four points in sorcery.

Defense does not really give anything that beneficial other than making Veigar a bit tankier. But, Veigar does not need to be that tanky because he will always want to be at the back of a team fight.

Utility: For utility, I upgrade my Flash and Teleport summoner spells as well as get the added maximum mana and the mana regen because veigar is mana dependent early game. The added movement speed will also help veigar in escaping.

The two spells I use:This spell is great in many ways. It can be used offensively to have veigar get in reach to use his stun or to use a baleful strike to a fleeing enemy with less than 10% health. Defensively it can really help as well. If someone suppresses you or any other cc, mash the flash button and since you have a little bit of survivability, you should be able to survive by getting out of reach of any other cc. Also can be used if you are being chased.Teleport is a very useful spell for veigar. First its amazing help in early game because you can get the items you need, regen health and mana and come back to mid, bot or top at full strength. It helps your team's map control by protecting against split pushes etc. Another great thing about teleporting with veigar, is that if you see a squishy champ like ashe overextending just a little, teleport to her, nuke and get a pretty easy kill.

Other ones you can use:Ghost is helpful for chasing and running away but flash is overall the better option for escaping cc. After all you should be able to outrun people just by using your event horizon.Exhaust is useful for eliminating any threats to you in team fights and helps for chasing a lot. Usually your teammates get this anyway, so I do not find this spell 100% necessary.

Veigar has three damage spells and one stun that can be used as an escape as well.
Passive: Equilibrium
Increases veigar's mana regen by .75 for each percent of mana missing. This is a nice passive that helps with your mana dependence

Q: Baleful Strike
This is one of your primary damage spell if not the best damage spell that you have. It has a very low cooldown making it spammable. What really makes this ability fantastic is its passive.
If you kill a minion with it, you permanently get 1 ability power. This will increase your ap in the long run by around 200+!
Also you get 1/2/3/4/5 ap for killing an enemy champion. Because of all these reasons is why I max this out first.
W: Dark Matter
Aside from your ult, this is your most powerful spell. You pick a location and a few seconds later dark matter falls from the sky highly damaging anyone in the area. This is amazing for farming and does insane amounts of aoe damage. What is also great about this is that it reveals the area you are aiming, so this prevents people from sneaking up on you from the bushes. It can also be used to steal dragons, and easily farm the wraiths and wolves. The only reason why I do not have this maxed first is that unless it is coupled with veigar's event horizon, it is very unreliable to hit.
E: Event Horizon
This spell has so much utility to it and is arguably one of the best stuns in the game. Veigar places a ring on the map that will stun anybody that passes through the perimeter. This spell can assure that dark matter and the rest of your combo can hit the other champion, and at the same time it can save you and your teammate's butt while trying to escape.
R: Primordial Burst
This ability is why veigar is called the anti-caster. This is the bane of champions like leblanc, brand, katarina, and many other ap carries. The ability works by having a set amount of damage plus 120% of veigar's ap, + 80% of the target's ap! This will always do at least 70% to the ap carry on the opposing team.

For beginners:

Starting Out:I start out with a meki pendant because it gives small mana regen per five which will help with veigar's mana dependency early game. It also build into a key item into your build, the tear of the goddess and eventually archangel's staff. The main reason why it is my first item is that it costs only 390 gold which lets you buy a health pot and mana pot, which will help you in lane. If you want, try out a doran ring first to help early game strength.

Early Game:As long as you have not gotten harassed enough to need to go back to your base. Wait until you have 955 gold. This will be spent on tear of the goddess, which will really help your mana starvation, and basic boots to help movement speed. Later then get sorcerer's shoes to get more magic pen and finally get catalyst the protector. The catalyst is one of my favorite items in the game because of its passive. It grants 250 health and 200 mana over 8 seconds on level up. This will give you the ability to harass without having to go back to base as often. This item is ideal for beginners because it helps them stay in lane

Quick Note: You may notice and be appalled that I do not have any ap items yet. But with my runes and farming with q, you should have around 80-100 ap.

Mid Game:With mid game you complete your core build and increase your ability power to about 300 if you have been farming well with baleful strike. The Rod of Ages gives you solid ap and mana, as well as a nice amount of health to help survivability just a little bit. This item is also a snowball item, which is why it is imperative you get it at around the 18-23 minute mark. The next item which will monumentally increase your ap is archangel's staff. This only gives 45 ap but its passives, will increase your ap way more than that. It adds three percent of your maximum mana as ap power. Since you already have a lot of mana from the tear as well as roa, this will greatly impact veigar's burst damage.

Late Game:Okay, this is where things really start to get fun. To start off late game, rush a rabadon's deathcap. This "deathly" item gives veigar 140 ap plus 30% increase in all ability power! With this item, veigar will have almost 700 ap as long as you keep farming baleful strike. After rabadon's deathcap, I get deathfire grasp. What is great about deathfire grasp is that it gives small mana regen which is nice, and also give 15% cooldown reduction. The best part about this spell is its active. The active states that the champion can deal magic damage to (30%+ 3.5% for every 100 ability power) of a target champions maximum health (60 second cooldown). This is great because it pretty much adds as another quick spell to use in your devastating combo. Lastly, I round off my build with a void staff. Void staff gives a whopping forty percent magic penetration, which will tear through a lot of the enemy's mr. I get it last because a lot of squishy champions get a banshee's veil as their last item, so this magic pen is not needed that much early or mid game.
*Quick note: I suggest you put deathfire grasp (if it isn't already) in the fifth slot. This is because it is easy to access with your index finger and when you can get to your ult quickly after using it.

For more advanced players:

At start:Veig is mana hungry early game so the first item he should start out with is meki pendant. It also allows you to get two health potions which helps you recover from harass. I do not suggest a mana potion because if you have meki pendant and the seals, you should be fine. After you get about 955 gold, go back to base after you push the lane out a little.

Early Game:After you go back to base, you should have a tear and boots of speed. This will solve your mana problems and give you movement speed to avoid other champion's attacks. After you have enough gold, you can then get kage's lucky pick and upgrade your boots. Kage's lucky pick gives you some small ap that helps your harass a little bit but also gives you a nice 5 gold per 10 sec to help you get other items. The boots gives you magic pen which tears through through some magic resistance.

Mid Game:
You are finally finishing an item build and this helps veigar tremendously. It gives cooldown reduction, mana regeneration, and gives you an amazing active. This item is a weaker version of morello's evil tome, but its active makes it the better option for veigar. The active gives you another spell that is a nuke that deals 30% +(3.5 per 100 ability power) of the target's current health in magic damage. Does a minimum of 200 damage. This gives veigar that extra spell to put away a champion. The archangel's gives you a boatload of ability power since you should have been farming your baleful strike and using your spells. This gives you around 200 ability power and a nice active.

Late game:The deathcap will give you a lot of ap to say the least. It is a staple in any ap carry and veigar is no exception here. The next item is zhonya's hourglass. This gives you a little bit of armor to survive a little bit, but its the active of this item that saves you. It puts you into a stasis for two seconds, making you untargetable and at the same time, not letting you move or cast spells. This helps in so many ways, have something on cd for two seconds, activate it, being focused by entire team activate it, tryndamere ult or karthus ult or any other aoe ult activate it. This increases you survivability by a lot. Lastly void staff gives you some more magic pen to get rid of any banshee's veil mr any other mr from any other items.

Conclusion: This gives you 900 ap in total around late game and makes veigar a beast with max cdr, magic pen, and two good actives. I do not see why you would not use this build!

The best place for veigar to play early game in my opinion is mid. Veigar needs to farm his q as best as he can as well as get to higher levels faster, and needs mid to do that. Veigar does get harassed very easily by almost any champ but if you are careful, you can still mid without feeding, or losing the first turret.
First always stay out of range of minions unless there are ones available to baleful strike. Sometimes you will even stun the opposing mid champion just to get a baleful strike in. As a result of this most of the opposing minions will come to your turret, dark matter the middle of the mob so they will always be one shotted by the turret, and the enemy cannot hurt the tower too much. If they are overextending to begin with, have a jungler or someone from bot or top gank to remind them to back off.
Although mid is probably the best option veigar can do really well in side lanes as long as he has a cooperative teammate the is tanky. Have your teammate try to stay back and just let you last hit a lot and stay away from the minions you are attacking. Propose a plan where you attack the caster minions while he attacks the melee minions for example.

When you get into a teamfight NEVER always be one of the first champions to go in. Veigar benefits from stunning everyone after the battle has started so that he can easily do damage without having to worry about being ccd. When attacking always aim your ult at an ap carry unless they are close to zero health, if that is the case aim for the ad carry.

All in all, veigar is a fragile champion but if used correctly can carry a team to victory. Please try the build or at least give it some thought before you vote. If you read the whole entire thing, thank you for your time. This is my first guide and I hope it is okay. Good or bad I would like some feedback.
Thanks again,
CJ2K