This is my first time attempting a build and I hope you like it. I started using Veigar when he was he free character and have been loving it ever since I have seen how good he is. Basically he is a nuke with feet and no one will stop you!

Pros:

Instant kill almost anyone!(The only one I ever seem to not be able to 1 hit is Cho' Gath)
Area of effect Stunning.(If you need to run, just pop in right on top of you and your good)

Cons:

Low health/Defense
Very mana heavy

Baleful Strike:

Great spell. I will get into it a little bit later but this is such a fantastic spell, if you are in a lane with minions, you should use it as much as possible. Its very good for minion killing because its got a low cooldown and low mana cost, and at the later game very deadly.

If your getting pressured by an enemy use this as a harassment ability and he will be scared to attack you, it is also a good kiting ability.

When ever this spell kills anything you increase your ap, and when you kill a champion with anything, your ap goes up again. Why wouldn't you love this spell?

Dark Matter:

Its a great spell in combination with the event horizon, but early game very mana heavy. Do not use it on minions until: You have a lot of mana. When it is at level 5 (you are level 14)
it will instantly kill all of the regular minions in the area.

Event Horizon:

The key part to the 1 hit combo!

It is an AOE stun that will save your life so many times. Like Dark Matter, extremely mana heavy early game, I would avoid using it unless you are either sure you can get the kill or you are trying to escape. Do not use waste this spell early game! A good tactic to use against champions is to cast this and when they are stunned drop the meteor. A wise trick for this is to cast it so that the champion is on the edge of it not in the middle, the spell will only stun them if they touch the edges.

Primordial Burst:

Your ulti, the most destructive thing you could possibly do to AP heavy champions. At level 3 (You are level 16, it does (500 + your ap + 80% of your opponents ap) which is ridiculous!
One thing though, do not use this unless you know you have the kill, do not waste it!

I'm just putting a little here at the bottom for the summoner abilities, Clarify is a great way for you to keep your mana high in the early game without you having to go back home and losing some of that crucial early exp.

Teleport can be swapped out for flash, that is personal preference.

If your going in with a full team you want to take middle by yourself, you can easily handle it trust me. You want to farm as many minions as you can early game because at the start your very easy to take down until around level 5 or 6. use your baleful strike to push your enemy back while you get some EXP.

Things not to do:

Use Dark Matter, Event Horizon, or Primordial Burst unless you have to.
Avoid direct confrontation with enemy champions. (If you see them just use a hit and run tactic)

I'm not going to go into a whole lot of detail over this,

Marks are to get over their magic resistance.
Glyph's, Seals, and Quints are for late game AP.

I picked the masteries the way I did to focus on increasing mana and reducing cooldowns for obvious reasons.

I'm not going to go into huge detail with the items,just going to focus on the important things you need to know.

Meki pendant: great for keeping your mana up early game.

Tear of the goddess: You want to get this as early as you can because with it, every time you cast a spell your total mana increases. Who wouldn't want that?

Deathfire's Grasp: The most important thing for you to get! The key component to your combo. I will be going over this in the next section.

Rabadon's Deathcap: It increases your AP so much it's unbelievable!

Using this combo you are almost able to 1 hit anything if you do it right. What you want to do is:

Use Event horizon and cast it so that the edge of your spell is just hitting the enemy.
As soon as you see it come up use your Dark matter.
Use your deathfire's grasp's Active ability (hotkey 3) to take away 30% of their total health! (If your fast enough, you can hit them with it before the meteor drops)
Use baleful strike to take away another chunk of their health.
If you think you can now kill them hit them with your ulti! If you hit them with everything listed above they should now be dead. (Cho' Gath is the only one I have not one hit using this combo)

So, for this combo to work you will need at least 605 mana if you are level 18 to use all of the spells. Just for an easy reference the hotkeys are:

E
W
Q
3 (if you bought the exact same items in the exact same order as I did)
R
Q (If it is recharged (If you followed the guide exactly it should), and somehow you did not kill them, hit em' again!

I hope you enjoyed the guide and it has helped you in some way, I have never gone negative using this build and am sure you will be able to do the same. Everyone will hate you when you drop your combo and they are instantly obliterated. I thank you for taking the time to read this and hope my build will help you as much as it helped me.