My very first guide... yaay~
After playing Veigar couple times, I noticed that AP items are giving me pretty "meh" ammount of AP. For a cost of some AP, I swapped into full rabbadon tanki Veigar once, and I met with my team hate, hating on me for building him like that. After scoring 20/7, they shuted up. Especially when I pressed R to kill Twitch...
Anyway, imagine Veigar chasing you, who have like 700 AP, 120 resistance and armor and 4000k HP. Pretty spooky, don't it?

Here we go... wall of text.
I used to take both Omen and Spirit because it's good for some reasons. First of all, Omen is greatest item vs ADC, just like Thorn. Omen decreases enemy ATT speed and crit damage, both are the most usual build for most ADC, while Spirit gives you pretty decent CD and makes you heal up near instantly with Warmong, but Banshee is the better option when enemy is bursting you up near instantly, or when they have Morgana.

Rabbadon is my core item on Veigar, as with passive it replaces most of AP items. In late game, when you farmed a bit, you should have around 700-800 AP, which is enought to one shot squishy, or pick easy penta with E+W when enemy is mid-low HP. Thanks to this item, I can turn veigar into literal rock - strudy as rock, painful like a rock. Steven would be proud of you, rock master, all you need is "Rockmaster veigar" to throw rocks at enemy with Q W and R... also, you can listen to rock. Why not?

Morellonomino... what ever it is called is good choice, as you are enought tanky to take on most of enemy damage and burst them up. It gives you amazing CD and ensures that you will never run out of mana.

Warmong gives you ability to face-dive at enemy, snare them, blow up and run away, waiting for your health to maximize again. Then you repeat until enemy is dead.

Zz'rot is good when your team struggles with pushing enemy lanes, also it gives pretty decent tankiness and turns you into Usain Bolt when towers are around... or their ruins. Tell Ziggs to blow them up eventually.

Banshee - take it when enemy Morgana ensures your death, or when Lux eats half of your HP with her combo. It's also better choice when you care more about surviving teamfights, where Spirit is less useful than Banshee.

Zhonya take it instead of Morellomonomi... screw it's name. This item will ensure that you'll survive when enemy team comes all out at your face, leaving them with lots of CD and you laughting at their HP bar dropping like a Skrillex bass.

Thornmail enemy ADC is butthurt. Take this. He'll beat himself up.

Hextech GLP is goot to slow enemy down and eat their health bar even more, but it's very situational and I'm not taking this item too often.

Hextech protobelt it's my favourite item to finish up enemy ADC when they survive my QW combo and my R is on CD. Also, lets you leave with ease when situation is against you. Take it when enemy is chasing you a lot, or when you don't have enought AP at the moment. Extra tip - eat minions, spam W at enemy.

Guardian Angel - Banshee is not enought to save you from enemy burst? Take GA, and when they burst you up, just wake up, E at enemy and collect kills with rest of your spells. Pretty complicated... no wait, it's not!

Red trinket - Teemo, Twitch, Eevelyn, Shaco etc.

Blue trinket - Choice for green trinket.

Green trinket - Choice for blue trinket. It's all about preferences.

This build is the same magic as with Nasus - all the same! Farm till late game, then come up and burst up enemy ADC or, in Veigar situation, whole enemy team.

Early game focus on minions. Q them all the time and watch you evil power rise, and when enemy jungler comes, E both enemies, burst them and leave till everything calms down. Then get back to farm. Alo cast W a lot at enemy, for extra AP from your passive. Go all in when enemy falls into your E, and cast everything you have at his face, you know, for free kill. But R can not hit enemy in time before W kills him... that's good I think.
During midgame enemies like to split push and go full team. When teamfight rises up,
go with E ASAP and let enemy die from your combo and your team. Such a sorcery... also don't try to tank up enemy team, you're not a real tank and this build only ensures that you will survive all the focus on you, letting you harvest free kills. Ah yes... enemy loves to focus Veigar, that's good that you're too tanky for them!
Late game ADC and AP is the most dangerous enemy. You saw all those Pentakills? Yeah, most of them are made by them, so when it comes to teamfight ALWAYS focus AP and ADC first unless you want your team to melt down... alongside with you. It's good that your E prevents enemy ADC from healing up... also, when you farmed a lot with Q'ing minions and W'ing enemies, you should have around 700 AP, 120 resist and armor and around 4K health at this point, so go now. Make enemy report you for being too god-like!