Hello summoners, I am a long time Kog and Veigar main. I recently noticed that a majority of popular Veigar builds are inefficient in the sense that they practically prohibit early game Q farming due to a zero mana build. So I've decided to make one!

PROS

Crazy burst
Infinite scaling
Highest single target burst in the game past 40min
AoE mega stun

CONS

Immobility
Bad early game
Needs peel and a team comp around him

You have to be wary when you play Veigar. Squishy jungle and hybrid bruiser top? Maybe not the best comp. CC bot support and Tank Jungle? That's the team comp. Most of the time your team composition requires a tank as a jungler and a tank/bruiser top. Though, a support CCBot like Leona and a tank jungler is good top. You simply need a frontline and a peeler. It's also best to hold off on playing Veigar unless you have a friend playing with you outside of the carry role.

Crystalline Flask with pots is the best starting build you can go. You rarely go anything else. The flask allows amazing sustain to keep up with the poking/trading meta mid laners. The mana from the flask and the pots allow you to Q farm until you can buy your chalice. By the time you run out of mana from flask and the pot you have around 1100 gold.
Doran's Ring with pots is rare. You really only do it if you end health I'm certain matchups. It doesn't supply enough mana if you constantly Q farm before chalice to never back until you can buy chalice. I only suggest it in a situation arises that requires it.

Chalice of Harmony is the key part of thus build. Chalice allows you to Q farm early game, which most builds don't allow you to do. The reason most builds don't have chalice is because they're looking for the early spike in damage that a DFG gives you. You might think that the early damage spike is needed, right? Nope. The chalice allows you to farm, which in turn allows you to be even MORE of a threats lategame. The idea is that chalice increases your damage lategame by sacrificing early damage.

This is where you Q farm constantly. That's legitimately the point of this phase for Veigar, farm farm farm! Now, that does not mean you should not fight your enemy. When you purchase your DFG (Which is not too far after your first back if you're farming well) you should be able to kill the enemy midlaner in 2 combos. First combo you must E->W->Q. That simple combo half healths a majority of midlaners. When they're half health your full combo can easily beat them. Before the heal back to full or back, do your full combo. E->W->DFG->R->Q->Ignite, instant kill. That process should be repeated several times through this phase while simultaneously farming. Though, giving up some Q farm to roam is worth it since your roam potential is outrageous.

At this point your farmed, have a bountiful amount of AP, and can 1shot everyone on their team. Your lack of mobility becomes a serious problem at this point. Your team is probably grouping and pushing, causing tension and team fights. In each team fight you must play it PERFECTLY.
Teamfights
Before the teamfight, stay behind your team. When they engage, don't be scared to hold off for a second or two, analyzing your approach. If you can land a stun of more than 2 people and at least 1 of those people is not a tank, use your stun. Disregard whether the enemy carries are their or not, it allows your team to steamroll their front line. Now, that oppurtunity is rare. A majority of the time, you don't use your spells and position your self on the outskirts of the team fight near their backline. Assuming your peeling, their tanks are too busy being assaulted by your carries and front line to attack you. That also have their cooldown down. Assuming you waited a few seconds for the fight to settle, their used a majority of their spells. This allows you a few seconds to kill their ADC/APC. Move in on them, then E->W-DFG->R->Q. Where'd their carry go?