Not Updated For Current Season
The masteries shown here are not yet updated for the current season, the guide author needs to set up the new masteries. As such, they will be different than the masteries you see in-game.
Veigar: Known to many as 'The glass cannon'. If you play him right, he becomes much more than a cannon. This Veigar build is meant for solo 1v1. Try to stay away from harass spells as they will eat you up early game and put you behind the curve.
Magic pen as much as possible, the nerf to the mana per 5 runes make them obsolete and unneeded with the meki pendent early. No magic pent seals so you can use those for mana per 5 (or AP if you like extra damage early).
Start with the pendent, it gives you the mana regen needed early game and you also have teleport, take advantage of if when you run out of mana, low on health and need a buy (try not to use it for just one, make it worth the trip back).
First back you should be able to at least pick up a Kage pick if not more. Just keep in mind that the CDR of the codex will help you farm your Q so if you can get that on your first back, screw the boots and get the CDR, you have flash to get you to tower for ganks and such.
After you get your DFG, you are ready to start shooting squishies up. If you can get a ganker to help you put a little damage on (and you have quick fingers) DFG's active will allow you to get kills if you can land your stuff. Stun+DM+DFG+ULT (quickly) will give you max dmg. After getting mejais, do whatever you can to kill the squishy and increase your stacks. Rylais helps your HP and gives you some slow with your quick q and the rest is obvious.
If being focused a lot by champions in teamfights, you may want to go with the Hourglass to preserve yourself until your tanks can come help you out.
I used to carry clarity flash and start with a tome but I find that the teleport is much more helpful for getting back to lane and keeping your q farmed up. Flash is not just a run away spell...it is a chase spell. If they are low but away from you, flash and land your stun on them and you win with basic spell combos. FLASH IS A MUST!
For solo 1v1, q is your best bet for direct damage. It instantly becomes a major harass after you level it to 4 and with items, it hits hard. DM is hard to land effectively, especially if they get Merc treads (reduced stun time) so just get the q and do work.
Farm your q! It is essential, it is easy to have over 100 ap right after you buy DFG, and that does an extra 3.5% of their health by just farming. DM scales 1:1 and your ult does even more. It should be easy to farm at least 60ish ap from your q by end game. After you get DFG, you can think more about farming champions, and those give you even more ap! =)
Veigar is squishy, always stay behind your tanks! Be weary of Fiddle's and Kassadins. Your biggest asset is to insta-kill a DPS champion (even sometimes if they have MR, you can insta-kill or remove them from fights). Yi takes a lot less damage if he is meditating so stun him or wait for him to be done and then do work!
Veigar's stun works best if you can get multiples, but the biggest contribution is taking out a DPS. Late game CDR is fairly low for Veigar (ult exception) so this allows you to still contribute even after you ult (DM again 1:1 ratio so make sure you land that...)
So, this has been a really quick guide but at the same time, maybe this may help you understand that Veigar is a very brutal champion. Be weary of teams with Banshees, as a team with all Banshees will practically eliminate Veigar, but give your phys dps a chance to gain an advantage as they will not have spend the 2700 on it. Go win!