I made this guide because I think Veigar is a great champ but there's just way to many players who can't play him like he is supposed to be played, so pls read this guid through it might just help your Veigar playing.
I main Veigar and I personaly think he needs tons of AP since he is to squishy anyways so I don't see the point in buying armor or magic resist since they're just gonna end up killing you anyway if you haven't killed them yet.

Pros:
Infinent ap
Ability to nuke everybody in late-game
Even if you lose early-game you will get back in mid/late-game

Cons
Extremly squishy
Early-game is a joke ("Is that a short joke?")
Has no gap-closer except Flash
Easy to counter with cc

Perfect:
Flash is a great escape toll and really nice if you are squishy.
Ignite: Great for getting kills before your burst can kill them in an instant.
Teleport: Good for when you roam and need to get back into lane to farm with Q .

Why not?:
Heal: Good if you are new to squishy champions and/or wanna sustain with only 2 health pots.
Barrier: Since Veigar is very squishy it might just save your life against champions with early burst and let you survive early-game.

Don't know:
Exhaust: If you are close enough to use this spell you shuld be able to just stun and nuke your opponent, but if you are against an assasin it can be usefull in stopping them from killing you before your E materialize.
Ghost: This spell can be used for escaping but that's what your Flash and E are for.
Clarity: If you want even more mana sustain then Clarity can work I guess.

Just DON'T:
Smite: I think you got lost somewhere, this is a season 6 guide for Veigar

Baleful Strike (Q):
THIS IS YOUR MOST IMPORTANT ABILITY. This spell could win you the game alone (if it was a 5 or more hours long game). This ability is mainly used for farming since it gives you +1 ap permanently for each minion you kill with this ability, siege minions gives +2. Also use this ability to kill your target since it also gives you +1/2/3/4/5 ap .
After It's nerf/buff you can now last-hit 2 minions at the same time or poke the enemy while farming. To do this line up 2 minions (or line up the enemy with a minion), then get the minion(s) low enough for you q to kill them and then hit them both with your q .

Dark Matter (W):
Except for your R this is your most damaging spell, but it's hard 2 hit with. So mostly use it on stunned enemies or just to try and poke. It is also a great way to clear minion waves quick. If you are being chased and don't have E your W may get them to flee or at least damage them a lot. Remember to aim in front of your enemies when fleeing so they'll run into your W , if they keep chasing.
Stacks 100% of your AP

Event Horizon (E):
This is the only spell you have that doesn't kill people itself, and your only build in escape. Event horizon is your only cc spell and it's therefor really usefull for making sure your enemies won't kill you first. After it's nerf it's gotten a lot harder to hit, but not impossible. When using E , just remember to place the edge of the hitbox where your enemy will be in 0,5 seconds. Leveling E up will stun the enemy for 0,25 seconds longer and lower it's cooldown.
Scales 0% with your ap

Primordial Burst (R):
Your most damaging spell. It is best against AP champions since it deals 80% of their AP in magic damage. But if you are fed it can alone give over 1500 magic dmg to the target without the enemy ap scaling.
Stacks 100% of your AP

The combo:
First you stun your target with E , then you quickly place your W on the stunned or about to be stunned target. If you are quick enough your target will be hit by your W before they are unstunned. Before your dark matter lands you'll also have time for an Q and even your ult if it'll kill them.

I first max QBalefull strike for the cooldown reduction so that you can farm alot with it, in level 5 it only has 4 seconds cooldown wich means that almost all your last-hits can be done with Q , and thats a lot of ap . Then I max W next for early burst since it in level 5 gives 320 magic damage plus all your ap[rabadon's deathcap], with a farmed Q , that should be about 700-800 magic damage (a lot). I personaly max E last. And of course I upgrade R as soon as possible.
If you are new to Veigar and his combo, some earlier points in his E is strongly recomended, since in level 1 it stuns only 0.3 seconds (totaly 1.5 seconds) longer than it takes your dark matter to land, (enough for me, but not for you, yet) but with each upgrade it stuns 0.25 seconds longer (level 5= 2.5 seconds). If that's what you wanna do then your skill sequence should mayby look like this:


If you wanna know how to use his abilities check the section called "Abilities" above.

In lane you will mostly try to avoid getting poked down while farming with your Q . Once you've gathered enough AP and have reached level 6, you can try and kill the enemy laner, IF you have enough damage (remember to do the math). If you don't have enough damage and aren't satisfied with just farming (which you should be), you can poke the enemy down with your QBalefull strike until they are low enough for you to stun and kill them. Another important thing to remember is, if your enemy fails a tower dive, make him pay for it, by stunning him under your tower and throw some Dark Matter in his/hers/its face.

One last thing to remember when playing as or with Veigar is that he can easliy be ganked by the enemy as he has only one escape toll when his flash is on cooldown, so save that E until necessary. Alos ward a lot so you can fall back before the jungler comes in for the kill. If you get ganked place your E in a way that stops the enemy laner and the jungler from getting to you.
It is also very easy for you to follow up on ganks from your own jungler. Just place your Event Horizon when your jungler ganks and use all you got togheter with your jungler. Unless you fail you'll AT LEAST get a Flash.

Ferocity:

Fury VS Sorcery:

Attack speed vs bonus damage on abilities , on Veigar? That's an esay one (Bonus damage on abilities)

Double Edged Sword VS Feast :

I'd chose Double Edged Sword any day on Veigar. You may be killing a lot of units with your Q , but you only heal 20 HP every 25 seconds. The only reason not to take Double Edged Sword is because you take 1,5% more damage, so if you feel like you die too much with Veigar you could try taking Feast .

Vampirism VS Natural Talent :

The spell vamp is nice for sustaining in lane, but in the end the bonus AP is just better on Veigar. Spell vamp helps you sustain in lane, AP helps with your burst and farming.

Bounty Hunter VS Oppressor :

Both are really good on Veigar. Oppressor let's you deal 2,5% more damage to anyone stunned by your Eevet horizon, but in the end after you killed at least 3 difrent enemy champions Bounty Hunter gives you more % damage even when the enemy is not cc'ed

Cunning:

Wanderer VS Savagery:

Savagery is useless for you in anyway possible. Why? Because you only got 1 single target spell (your ult ) and you can't (and shouldn't use it on minions. Because of this there is only one choice left, Wanderer , plus the movement speed can help you roam.

Runic Affinity VS Secret Stash VS Assassin :

On Veigar these 3 keystones really shows the point of these new masteries. All of these are good on Veigar and therefor I won't tell you wich to chose, but which playstyle each of them represent.
Take Runic Affinity if you expect to get and want to get blue buff.
Secret Stash will help you sustain even more in lane, both with manatear of goddess and HP .
Last but defiently not least the new Assassin . Take this if you wanna give more damage in lane, both to minions and champions , and if you wanna 1v1 (or mayby even 1v5) them all.

Merciless VS Meditation :

Both are really good on Veigar. Meditation gives great manatear of goddess sustain in lane, while Merciless helps you finish of damaged targets (can really help on those tanks that belive they can't die to your burst)

Precision VS Intelligence :

Precision is good if the enemy have heavy 2 or 3 tanks , but in general the Cooldown reduction you get from Intelligence will help you stack a LOT of AP for late-game

Stormrager's Surge VS Thunderlord's Decree VS Windspeaker's Blessing :

So here there is pretty much only 1 choice and that is Thunderlord's Decree . You have no sheilds or heals for Windspeaker's Blessing and while Stormrager's Surge can help you chase down targets that survive your burst , but they should already be dead (especially with Thunderlord's Decree ). Plus Thunderlord's Decree helps you win trades with Q + W if you thorw in an basic attack and your full combo (Q + W + R) also procs Thunderlord's Decree

Resolve:

The Resolve Tree is mostly/almost only for tanks or tanky champions, therefor it's not usefull on Veigar

In general I prefer to go with instant AP Glyphs and Quints, combined with magic penetration marks, since it in the ende will give you more damage than instant AP marks, and for seals you should either pick scaling AP for damage or Instant HP to make you a little more tanky, like this:
But if you fear ending up vs Talon or Zed (either because they become Meta, or it's a ranked game where they counterpick you) then you'll wanna take some armor Seals, rather than AP , like this:
If you fell confident and would like to leterally 1-hit your opponents (without 1000 stacks on your Q ) then you can go with full scaling AP Seals, Glyphs and Quints (do this at your own risk):

Items
Counterpicks