Veigar is a somewhat strange character in League of Legends. He's an infinitely-scaling mage with crazy utility, a kit as easy to use as a coloring book, and, at the time of writing (12.2), is still not considered S+ tier on u.gg in any role. I disagree with this tierlist, and hopefully can convince you to disagree as well by the end of this guide.

Veigar's basic kit includes 3 abilites, an ultimate, and a passive, just like most other champions. Each ability is pretty simple, so these descriptions will be short and sweet.

Passive

Phenomenal Evil Power is Veigar's passive ability. Each champion struck with a spell from Veigar grants a stack, each takedown grants 5 stacks, and each stack permanently increases your ability power by one.

Q Ability

Baleful Strike was just buffed in patch 12.2. This ability deals magic damage to the first two enemies hit. Any small minion killed by this ability grants one stack of Phenomenal Evil Power, and any large minion or large jungle monster killed by this ability grants three stacks of Phenomenal Evil Power. This ability is your main minion-killing tool, as it gives the easiest access to Veigar's infinite scaling.

W Ability

Dark Matter is Veigar's W ability. The ability summons what is effectively a magical airstrike, landing a short time after it is casted. Stacks of Phenomenal Evil Power reduce Dark Matter's cooldown.

E Ability

Event Horizon, also simply known as "the cage," is Veigar's main utility and playmaking ability. It summons a circular cage that takes 0.5 seconds to appear and lasts for 3 seconds, stunning anyone who touches the walls of the cage.

Ultimate Ability

Primordial Burst is Veigar's ultimate ability, and may actually be the simplest ultimate in the entire game. When you press R, Veigar summons a big magical ball that does a lot of damage.

Minion last hitting is difficult for any new player, and Veigar's low earlygame damage makes it even harder. Here are some basic tips to help you CS with Veigar's Baleful Strike and scale even quicker.

• Left Clicking

  Left clicking on minions will show you their exact health in the top-left of your screen. This information, combined with a simple damage test, can help you get that early CS and increase your ability power.
  
  

• Lining up minions

  Baleful Strike can hit two enemies. Using this knowledge, we can try to set up opportunities in laning phase by positioning ourselves so Baleful Strike can kill a minion before piercing through to kill another. Doing this on a cannon minion and a normal minion will grant 4 stacks of Phenomenal Evil Power.
  
  

• Auto Attacking

  Your earlygame auto attacks on Veigar are incredibly weak. You can use this to your advantage, however, as you can lightly poke minions to be able to set them up to be last hit by Baleful Strike

Teamfights are where Veigar strays away from the typical botlane meta. You're actually not super squishy. I personally don't think I've been one tapped by an Akali or a Talon yet, even though most other hyperscaling characters would be. You can be and want to be a bit of a middle ground in your team. Playing ahead of your support is certaintly possible, but make sure you have enough cc off cooldown or a flash to be able to escape.

Your main goal is to be trapping their tanks and bruisers in your Event Horizon, following up with a Dark Matter cast and an Everfrost activation, then quickly following that with a Baleful Strike and Primordial Burst. This kills or almost kills about 95% of the champions in this game. This is your basic combo and stuns enemies for a minimum of 3 seconds, up to a maximum of 5 seconds of stun.

E-W-Everfrost-Q-R

Event Horizon -> Dark Matter -> Everfrost -> Baleful Strike -> Primordial Burst

E-W-Q-R

Event Horizon -> Dark Matter -> Baleful Strike -> Primordial Burst

E-W-Q

Event Horizon -> Dark Matter -> Baleful Strike

CC-E-W-Q

Friendly CC -> Event Horizon -> Dark Matter -> Baleful Strike

Your Dark Matter has a small delay between casting and dealing damage. League of Legends has no invincibility frames, so feel free to throw other projectiles before it lands.

Poking in lane should only be done with Baleful Strike, as using Dark Matter to poke will result in you running out of mana before getting Everfrost.

If the enemy team has a full Ability Power (AP) focused bot lane matchup, take the Magic Resistance shard. If it's full Attack Damage (AD), take the Armor shard. If it's a mixed matchup in botlane, take the Health shard.

Feel free to ignore early teamfights. You are almost entirely useless in them before you get Everfrost and Sorcerer's Shoes. Of course, if you're safe to help, then do so.

Make sure to buy Control Wards often. Don't take a control ward over an item completion or a Lost Chapter, but pick up as many as you have the spare gold for during the game.