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.
So, nothing unnecessary here - no spell introduction and etc., only what is needed. I'll try to explain, mainly, everything i do, and what you can't find in other guides.
As you can see, it goes one to balefull, second to horizont event and third one to balefull again. Why so, you may ask. Balefull strike's passive is pretty nice, so two points in it in early game will allow you to farm creeps and gain those extra AP points fro later game. After those, i mainly level up my Dark matter, which makes this build extremely useufll and powerfull. Primordial burst at your own wish - personally i level it up at lv 6 skipping Dark matter. It is not recommended, but im hasty and i like to do kills fast. Leveling dark matter early allows you to earn money earlier, let's you push enemy champions better and easier, so in the end, you might have less kills, but will have more money, than others, while primordial burst allows you for early game kills.
Flash - can be played either defensively or offensively. I dont need to explain anymore how to use it.
Teleport - with this build it is mainly recommended to go mid, so going for fast shopping and getting back is nothing better for you, and nothing worse for yuor enemy. It also gives you a chance to catch up with your friends in a team fight, where Veigar(all may think different) is VERY usefull.
Well, runes can vary on your choice - mainly, some AP for the beggining that your dark matter would kill back lane minnions, some penetrion for usefulness throughout whole game. But they can vary on your choice, i'll repeat myself.
Nothing much to tell - standart caster/AP masteries.
This is where this build of items shows that it is better than others(In my opinion ofcourse)
Start game with Meki's pendant and 2 health potions.
When you have enough money, or, if you were forced to go back, buy Chalice fo Harmony and Ruby crystal. They are very important to get as soon as possible. Chalice is usefull even after the new Veigar patch, which will give you more and more mana regen. that allows you to spam a lot dark matter, making you a equal to best pushers in game. Ruby crystal is for additional survivability, that u will be lacking in early game, before you max out your dark matter. Sell it whenever you need a free space for better items(optional - after soul shroud/before Zhonya's ring).
Well, later you just build rest of the items:
After getting these items, you mgiht consider selling your Chalice of Harmony/Soul shroud and getting Frozen heart and/or Rylai's scepter.
When you get your Zhonya's Ring, it should be a must that on your every journey home, you would buy an Elixir of Brilliance. It gives you a decent ammount of AP and reduces your CD's, allowing you to cast Primrodial burst, which at this point shoudl hit for 1k++, even more ofter, combined with Soul Shroud/Frozen Heart and Golem buff.
Play defensively. Very defensively. All your job is to last hit minnions wit hyour balefull strike, building some AP for your Dark Matter. Recall when you have enough money for challice and ruby.
When you max out(or even at lv4) you should be able to one hit back vawe of creeps - DO IT. It will earn you loads of money, allowing you to buy items even if you died a lot and didnt got much kills/assists(as mentioned above).
Mid game starts when you buy your Mejai's soulstealer. Then it's time roam a bit, getting free kills with your combo of dark blaefull+matter+horizont+primordial burst(if needed at all)+balefull. Just do not forget, that your balefull awards you extra AP for killing blow with it.
Do not hesitate to use your teleport to get back to your lane, or to help your friends.
Same as Mid game, just do not roam alone anymore, move with team, do not engage first. Use your Horizont Event to withdraw from battlefield, help with it's stun to hel pyour friend flee too, but dont risk too much. Buy Brilliance Pots.
Anyway. Everything here is by me, sicne a lot of Veigar's i see are stupid, run out of mana SUPER fast(what with this build will never happen).
But everything must be used with brains.
Veigar is Hard class, you must practice a lot to become good. And though, it is not considered usefull in ranked matches, so it is your call, to choose it as your main or not.
P.S. there mgith be some grammar mistakes.
P.S.S. Leave comments with your oppinions. I will update my guide with yuor shotu outs or if i will find something by myself.
Hope it helps you a little bit more, than other guides.