I love AP champions and Veigar is simply one of the most powerful AP champions in the game. Naturally, he is my favorite champion.
1. Champion Basics
Veigar is a burst anti-caster caster. His first ability is Baleful Strike. He launches a small ball of energy toward a target enemy and deals magic damage. This is your basic nuke though it has a secondary effect and a passive. If a target dies to a Baleful Strike, Veigar gains 1 AP and if Veigar slays an enemy champion, he gains 1/2/3/4/5 AP (depending on the level of Baleful Strike). This ability should be used as often as possible to last hit minions. Early game it is rather weak so I don't bother to harass enemy champions until it is level 3.
His next ability is Dark Matter. Veigar targets an area on the ground and summons a meteor of dark matter to land after a short delay. It takes about 1.2 seconds from the cast of dark matter to the meteor crashing into the ground, this means you have to aim it. The way around having people move out of it is to use this with Veigars next ability, Event Horizon. This ability scales 1:1 with AP and hits very hard.
Event Horizon is Veigars last basic ability and is, in my opinion, the most powerful and potent. Veigar creates a medium sized cage at a location and enemy targets passing through the cage limits are stunned for 1.5/1.75/2.25/2.5. You can drop the cage limits directly on an enemy champion, forcing an immediate stun which can be followed by a sure-fire dark matter. This combination is extremely deadly later in the game.
Veigars ultimate ability is Primordial Burst. Primordial burst launches a large burst of anti-energy towards an enemy champion, dealing large base damage, plus Veigars AP, plus the enemy champions AP in damage. Now knowing that, you would think the best is the use this on enemy casters. While this is technically correct, Veigar hits so hard that many times, any target is viable because you just overkill casters anyways.
Equilibrium is Veigars passive and it increases mana regen by 0.75% per 1% of mana he's missing, identical to Chalice of Harmony.
2. Champion Set-up
There are several ways to play Veigar. I prefer to build him to his weaknesses with my pre-game set-up and build to his strengths in my in-game items.
For runes I take Magic pen marks, health per level seals, mana regen per level glyphs and straight AP quints.
For masteries I take a standard 9/0/21 build, picking up improved buff durations and improved flash.
I take ignite and flash for summoner skills, though ghost is a solid choice as well.
3. Item Build
My play style with Veigar promotes a rather odd build but because of his insane amount of control with his Event Horizon. I build almost entirely damage with very little regard for my own defenses.
I start the game with a meki pendant and 2 health potions. This allows for high usage of Q stacking AP very quickly. I cannot stress this enough, to make this hero powerful, you must learn to last hit effectively with Q. Any mobs you cannot last hit with Baleful, you must attempt to last hit with your standard attack. This hero snowballs very quickly out of control if you have a good start so you must learn to do everything in your power to have a strong lane and make a lot of money quickly.
Depending on your situation in the lane, I like to rush chalice, as the mana regen allows you to play very aggressive with the combination of stun control and Baleful harass. Next up is Sorcerer's shoes. The magic pen plays nicely, and you're up to 28.5 base magic pen, turning whoever doesn't have any MR into an incredibly easy target.
Now the next item comes down to choice and play style. I pick up Mejai's because Veigar thrives on kills and can pick them up very easily. If you get a Mejai's and pick up a couple kills early, your AP will skyrocket so high that enemy champions will be forced to get MR just to live through your stun. The other choice is just straight rushing a Rabadon's Deathcap, which is your item after Mejai's anyways. Try both, see which play style fits you.
After the Rabadon's Deathcap, take a look at there team and look for two things. Is there a lot of physical damage that you can't avoid and how much magic resist are they getting. If they can't get to you, get the void staff and continue destroying kids in seconds. If they have champions like Pantheon and Caitlyn, I'd suggest getting your Zhonya's Hourglass first however.
At this point, you should have a Chalice, Sorc boots, Rabadons and Void staff. You may also have Mejai's and Zhonya's. If the game lasts this long and you don't have inventory space, sell your chalice and pick up an Archangel's Rod. If you aren't using your Mejai's as much as you should be (over 10 stacks) sell it and pick up a Deathfire Grasp.
A note on the Deathfire Grasp by the way. You can pick this up instead of the Mejai's early game if you are pro-active in using it. It is also very useful against Cho'gath / Vladimir.
4. Skill Order
My traditional skill order is as follows:
1. Q (Baleful Strike) 11. R (Primordial Burst)
2. E (Event Horizon) 12. E (Event Horizon)
3. Q (Baleful Strike) 13. E (Event Horizon)
4. W (Dark Matter) 14. W (Dark Matter)
5. Q (Baleful Strike) 15. W (Dark Matter)
6. R (Primordial Burst) 16. R (Primordial Burst)
7. Q (Baleful Strike) 17. W (Dark Matter)
8. E (Event Horizon) 18. W (Dark Matter)
9. Q (Baleful Strike)
10. E (Event Horizon)
You may find you want more damage or have a lot of team stuns and don't need yours in which case you can get more W and less E. Though for the most part the E is the single most broken spell in the game currently so I like to max it earlier.
5. Playing Veigar: The early game
Soloing on Veigar is very awkward. Most solo champions can hold there own in the first couple levels but Veigar is at the mercy of his opponent. Learn to read your opponents. Are they zoning you out? Are they playing passively? Are they playing defensively? These are extremely important to notice. A passive opponent allows you to get a lot of last hits with your melee as well as your Q. A defensive opponent will usually only attack you if you attack them. If this is the case, ignore them until level 5. At level 5 you have the tools to outplay most champions. If you are getting zoned out, there isn't anything you can do. Sit at your tower and do not die. Grab last hits whenever possible and call for a gank.
If you are playing in a duo lane, reading your opponents is just as important, but reading your ally is also important. At leveling three, you can do decent damage with Q and your stun will lock someone down, allowing for very easy first bloods.
6. The All-Smash
One the greatest features of Veigar is that he can catch someone in an incredibly long ranged stun, run up, instantly kill them and run off and be completely safe. If you can't nab the kill, just don't use your ultimate. Save your ultimate for secure kills and use ignite whenever you go for a kill. Knocking an enemy out of the fight instantly is so useful for your team so kill whenever you can.
The All-Smash combo is as follows. Stun -> Dark Matter -> Baleful Strike -> Primordial Burst -> Ignite
At 500 AP, this combo will do about 2000 magic damage on top of the true damage of ignite. 500 AP is even low for Veigar though. You'll have up to 1200 AP some games, and maybe even higher. A full Veigar combo at 1200 AP will smash a non-AP target for upwards of 4000 damage.
If you have any questions about my Veigar build, please put them in the comments and I'll do my best to answer them. I should also point out that I have been streaming my games recently so I might put the stream URL somewhere if you guys are interested. Also, this is how I play him, you may have other strategy's which work and I encourage you to make any adjustments you need to to make Veigar work for you.