Welcome to my seemingly unconventional guide to one of the highest damage dealers in League of Legends: Veigar.
I started playing League in season 2 and picked up Veigar in season 4. He hasn't seen many huge changes but his kit has been slightly altered (for better or for worse)
Here is my op.gg I've been trying to one trick Veigar in ranked and it got me to Plat 5 this season mostly carrying every game I won. I was never serious about ranked until this season because I wasn't confident in any one champion until recently when I found this build, and how good it can be. Props to Screaming (the #1 Veigar player in NA) for this build. I have a 67% win rate with him in 112 games this season. It would be higher but I discovered this insane build about 60 games in.
So here's my guide to one-shotting most champions and tilting players along the way.

This is a pretty different build than the standard build that most Veigar players run.
The most notable differences are:
NO
First item morellonomicon
Thunderlords
I can explain why I don't run these but first let's look at what the replacements are:
REPLACE WITH
Hextech GLP-800
Stormraiders

DEFENSE FOR GLP-800
So let me explain why I choose to go Hextech GLP-800 instead of morellos. I noticed that the early game was always difficult for Veigar, and I also noticed that I wasn't focusing on trying to kill people that often. So why not just get extra health, mana, and essentially another spell.

Some people might think that the early game damage is traded for survivability, but when you realize that GLP scales with AP and Veigar scales infinitely, you start to notice that GLP is just another spell in your rotation.
GLP can also be used defensively as a way to run away from threats. It can be used to slow enemies that are running away, and it allows you to stun them.

Morellos gives you 20% CDR which is very helpful, but I get CDR boots, intelligence mastery, and CDR runes. This allows me to have a suitable amount of CDR by the time I complete tier 2 boots.

DEFENSE FOR STORMRAIDERS
Back when I ran a standard build, thunderlord's helped me secure so many kills that I never thought I would use another keystone. However, this build is focused on surviving and dealing tons of damage, so the most optimal choice is stormraiders.
Stormraiders allows you gain huge bursts of move speed when you deal 30% max health to someone in a short period of time. This is very easy to do with Veigar.
Picture yourself getting ganked; all you need to do is put down your E for escape, Q + W (sometimes even Q will proc it), and then you'll have the move speed to escape.
This has saved me so many times in the past.
Another way it is useful is when you're in a teamfighting. You can easily deal 30% damage to someone in a team fight, and this allows you to stay mobile and hard to catch.

Well, Veigar has a couple of fun features that only a few champions will ever have: he infinitely scales. This, coupled with low cooldown and survivability items can make Veigar almost impossible to deal with if played well.
The most satisfying part about playing Veigar is one shotting people with your rotation. It's almost as satisfying as the enemy team raging and claiming that Veigar is OP.
He doesn't take much skill to play proficiently but mastering his stuns and positioning will take some time.
Let's see the pros/cons.

PROS:
+Highest burst damage in the game
+Low cdr = spamable E
+AOE 2.5 second stun at level 5 E
+Hard to kill with this build
+Very good when ahead
So let me elaborate on the pros.
1) His burst is unmatched and since he has a very low pick rate, many people underestimate his damage.
2) Once you get +35% CDR, you will be able to E twice in a team fight and maybe even more if the fight is slow.
3)His E is one of the best spells in the game. In the late game, if anyone gets caught in it, they will most likely die even if they don't get stunned. (Unless they have flash or an ability that avoids it)
4) This build lets him play aggro and safe when he needs to. In team fights, you can stay in the back and unleash burst damage and stuns while being perfectly safe due to increased health and banshees.
5) If you get a lead and play smart, then you can easily carry the game with constant roams and lane pressure.

-Terrible early game
-Terrible early game farming
-Easily exploitable due to no escapes
-If behind, completely usless
-Many bad matchups
Lets elaborate again.
1) His early game is downright garbage due to his high cooldowns and low damage. That is why you need to play for the late game.
2) If your lane opponent has good wave clear you may miss many CS under the tower.
3) Even though his E can be defensively, if the jungler camps you then it will be hard to recover unless they make mistakes.
4) If you get put behind, it will be hard to catch up with this build. This build sacrifices early game damage from lost chapter and codex, thus a behind Veigar with this build will just be a walking stun. Sometimes the mid game will be a fight for CS with your teammates and this puts you even further behind.
5) As noted in the matchups up above, there are a lot of hard champs to lane against. The smartest thing to do is just to play safe and farm until you get confident with Veigar's damage and can make plays based on that knowledge.

Vegiar gets free stacks of AP for landing an ability on someone, getting a kill/assist, and farming with your Q.
Here are the values:
1 Stack
- Hitting a champion with any ability
- Killing a minion with Q
- Killing a champ with Q

2 Stacks
-Killing a large minion or monster with Q

5 Stacks
-Getting a kill or an assist

It is very important to fight a lot as Veigar because you need stacks to thrive.
In a low ELO game where fighting constantly is commonplace, getting 300 stacks by 25 minutes isn't that hard.

The standard amount of stacks you want by 20 minutes is 200.
If you have more, great! You are doing well and will probably win that game.
If you have less, refer to the values below

STACKS AT 20 MINUTES
>= 200 - Fantastic

150-199 - Not bad at all, this is average and will do just fine.

100-149 - It's obvious that you are behind or just not good at stacking. Upper ends of 140 aren't terrible but you have to catch up.

50-99 -Ouch, things are going very badly and you will not have the damage to contribute much to your team. Farm it out or build utility

Any lower and I'm going to assume that you are supporting.

Well, I hope you enjoyed my guide and you join the hundreds of players that main Veigar. He has very apparent problems, but in the right hands, he is very useful to your team.
Do your best to be the bursty anti-fun champion that everyone should fear.
Good Luck! Thanks for checking out my guide!