This is the Veigar build I am currently using, and it will be precise in telling you my strategy to carry. It focuses on building magic penetration as a benefit of stacking additional AP from Veigar's Q.

This is the most important thing you need to understand about Veigar.

Your Q stacks ability power, and can hit two enemies.

If you kill a creep with it, you gain 1AP.
If you kill a cannon, you gain 2AP.
If you kill a champ, you gain 5AP.
If you assist a kill on a champ, you gain 5AP.

During laning phase, focus on last hitting with your q. Auto minions that are close together, and then q them together to gain 2AP. Lining up a cannon and a creep with q will give you 3AP! It is an essential mana saver and critical to playing Veigar to his top potential.

If a champion gets close, try to hit them AND finish a minion. The main point of laning phase with Veigar is to get a much AP from farming as possible. Make this your top priority.

Once you have these two items, you can confidently attack the other mid-laner.

No matter what ability they use, your q will out-trade them if you have been stacking AP correctly.

Typically when I get needlessly large rod, I will have over 130AP. The opposing champ will have ~70. See the difference in damage for yourself.

E Cage ---> W dark matter ---> Q baleful strike

This is your bread and butter.

If a champion is pissing you off, Press R.

Proceed to laugh.

Veigar's abilities are great for teamfights. Here are some things I like to do.

1. Attack a turret with teammates and throw up a cage around or behind it. The other team will be stunned or zoned, and then your team will instinctively attack them.

2. Ult + Q an adc to instantly kill them and take advantage in a team fight.

3. Drop dark matter w's like mines when being chased by the enemy team. They will typically walk right into them, realize you are huge, then run away. If you land a Q you will kill them.

4. Throw up the e cage to stop assassins and other big play champs from getting the glory.

5. Zone champs away from baron and dragon with cage.

6. Stun and group creeps with a cage and then use dark matter on the group.

7. [my favorite] Late game 1-shot then enemy mage with your ult. The Katarina with max stacks will all-chat "nerf veigar", not understanding that your ult scales with their AP.

This scales with your opponent's AP. Any mage in the game will take a ton of damage because of this, regardless if they are fed. Remember that you are always useful as Veigar.

Pros:

1. Over 1000AP late game. Arguably the best late-game mage in the game.

2. Amazing CC with the cage
3. Amazing Damage
4. IMO best scaling mage because of AP stacking. I am always ahead in AP for every game I play.

Cons:

1. Squishy. However, this can be countered with good q stacking, as well as the shield from seraph, health from liandry's, and health and MR from banshee's veil. A late game yasuo or a fed ADC is a caution.

2. Immobile. Your cage is your best friend for this. If you miss, your only chance is to all in whoever is chasing you to kill them first.

3. Mana-Hog. Early game you are always low. After tear is stacked, you are fine. Veigar also gains more mana the less that he has.

Magic penetration helps you scale like a monster through the game, especially late. Your nuking potential is much higher, and you can actually stop champions like darius, garen, and Riven that spiral out of control and would otherwise be a nightmare for your team.