He has the highest potential AP of any Champion and likely will have the highest AP in any game he's playing in.

His Dark Matter can easily take out 700-800 damage chunks out of groups of enemies every 10 seconds and is really good at clearing creep waves to keep him gold capped and the enemy team pressured during pushes.

Primordial Burst against enemy mages can do upwards 1400 Damage

Event Horizon can be used to stop enemy pushes and separate enemy teams and mess up initiations.

Baleful Strike is only really useful for increasing your AP, though spam harassing with a 500+ damage nuke every 4 seconds can definitely help your poke game. Range is bad though.

Build-wise you're probably back line but you need to close distance fast after initiation to get off a good EH so if you can get away with it Boots of Swiftness over Mercury Treads. Veigar's got great HP growth so he's not particularly squishy for a caster but you're still going to benefit from a few sturdy items. My typical build order is:

Chalice
Boots of Swiftness
Mejai's * if doing well early enough, otherwise just save
Rylai's * having this makes your 4 second Baleful a very reliable snare to help chase/secure kills
RoA * get this before Rylai's if you feel the early Catalyst will help your survival
Zhonya's * the passive boost to AP is huge on a champ that has the highest AP of anyone

Depending on whether or not you snowballed your Mejai's you may have an empty slot that will let you stack a good defensive item. Banshee's is okay if you're getting disabled though you should really be in the back line dropping Event Horizon's and Dark Matter's into skirmishes making you a hard target to hit with disables. Otherwise, Thornmail to discourage that aggressive melee carry is a great choice as well.

Veigar's strengths come from being in and out of combat from a distance and anytime you can abuse Fog of War, ramps, distance, or positioning to get shots off on enemies without them being able to retaliate against you is a good thing.

For that reason, I'm more fond of items that discourage people to focus you for that last defensive slot than something that will make you more tanky and let you get into the fray, that's exactly what you don't want to do.

Void Staff can be taken if it becomes clear that they're going to stack Magic Resistance.

Summoner Spells, Chalice is far better than Clarity but Clarity will let you build up your AP early on in the laning phase with Baneful and becomes an entirely support spell mid-late game for your team. Veigar could do worse than take Clarity but it's not on my high list.

Ghost is quite useful, as both an escape and to help your positioning during team fights and skirmishes as well as making you very adept at cleaning up with Event Horizon and Rylai's.

Teleport is always good since Veigar is such a strong anti-push hero, being able to rapidly defend and push back large creep waves for gold is useful.

Not a fan of Heal, Flash, Exhaust, Smite, Revive, etc.

Ignite and Fortify have their uses but other heroes are better suited to carry them.

I've never had Mana problems after getting Chalice. I even took Clarity a few games because I was uncertain how hard I could spam and stopped using it midway through the game because I rarely had mana problems. It can be rough early on, but there just really aren't any good Mana Regeneration items outside of Chalice for the cost. You can of course, if it gets to that point substitute the Chalice for an Archangel's Staff to max out your itemslots.

I find core is definitely:

Boots of Swiftness / Mercury Treads
Chalice
Rylai's Scepter
Zhonya's Ring

But you'll likely benefit from an early RoA or Void Staff depending on the way the game is going and a lategame defensive item makes more sense to further discourage people from focusing you and allowing you to output more aoe damage.

Only in the very late stages where I'm rolling in elixirs and gold would I consider adopting an Archangel's Staff. I feel like the staff gives 3 things: Mana, Mana Regen, Ability Power. For the cost, I find a Chalice does the first two fairly well and that Rylai's/Zhonya's have more utility in the midgame for Veigar and a good defensive item will take you a long way before you get to an Archangel's.

If you have a team that's running Soul Shroud or packing Clarity you can opt for an early Tears instead of Chalice and carry that into an Archangel's a little sooner. That could be an alternate route that I'll have to try out.

Dream you didn't read this guy right, he is saying you dont need chalice, his passive is a weak chalice now and also he doesnt snowball ap from skills like he use to unless he is rolling the other team.

One of the best tricks with veiger is E positioning, if running you cast it were they have to run through 2 walls of your stun if they have a banshees or cleanse...same goes for catching someone.

You need to learn to get more than one person on the edge of EH.

Also, early on it's better to pump DM than EH. Get one point in EH at level two and pump baleful and DM until later game.

I like to start with a meki pendant, then hopefully I don't have to recall until I can get kage's pick and lvl 1 boots.

Then I like to go for the mana teardrop (name escapes me). This is followed by finishing the deathfire grasp. After that, I go for rod of ages for hp and mana and ap, then finish archangel staff for mana and ap. This is followed by a zhonya's ring.

Boots
Deathfire Grasp
Rod of Ages (or Rylai's, but if I can get it early enough I go RoA)
Archangel Staff
Zhonya's Ring
-your choice

I typically don't get to 6 items in game. It's low AP early, but more mana so you can keep farming and building your AP with bale (Q). Then the AP starts to pick up, and you can start picking apart folk with stun-deathfire-darkmatter-primordial-bale combos.

Again, mine is more mana early, but that allows me to farm AP on minions in the lane.

Buy 5 archangels staffs and boots and then lol @ 1k+ap.