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.
Things to keep in mind when playing Veigar
Veigar should take mid lane for a fast leveling since he is kinda weak in early game.
Play defensive while soloing mid and always keep an eye on the mini map to avoid getting ganked.
Try last hitting minions with baleful strike as often as you can.
Get the blue buff whenever you can also, that will allow you to spam spells making your life a lot easier.
When you get your Deathfire Grasp is complete, things start getting interesting. That's most likely when you'll start being able to throw enemie champs from full hp to death in about 2 secs with the combo EW2RQ (since I usualy set Deathfire Grasp on the 2nd inventory slot, I use "2" as a short cut to it).
After that, it's easymode. Just follow the skills and items build and enjoy you nukes.