Veigar, AP Powerhouse
I remember back when I first started playing League of Legends my first champ that I used was Veigar. He was free and had a remarkable (and I'm sure coincidental) resemblance to Vivi from FFVIIII, and he was and is my favorite champion (sorry, Cho'Gath). Yet whenever I played as Veigar it was just an arms race with the other team's AP champs, just madly grabbing at any AP boost I could get. I always made sure to last hit with Baleful Strike to get permanent AP boosts, I would always max out Dark Matter to get the most AP damage out at once, and I'd always save my ult for the other team's main source of AP damage (since it does bonus damage based on the target's AP). League of Legends was prone to getting stale then, but everything is different now.
Same Champ, New Feel
With the new shop and the new items that came with it, Veigar feels like a new champ. Instead of madly rushing for AP I can see some of Veigar's major flaws that I didn't really notice before. I used to assume that Veigar's passive would balance his mana consumption and regeneration... but really it only keeps you from always being out of mana. Now, I know to combine Frozen Heart and Tear of The Goddess. Sure a Rabadon's Deathcap and some high AP boosting items (like a Rod of Ages or Zhonya's Hourglass) is a great combo, but Tear of The Goddess is absolutely necessary if you want to stay in fights. Really every stage of ToTG is great for Veigar, since his passive increases mana regen per every 1% of mana missing, and the next stage (Archangel's Staff) gives you an AP boost based on your total mana as well as a flat rate. So instead of rushing for Deathfire's Grasp as a necessary item, I'm now building ToTG through it's stages can build Frozen Heart and Rod of Ages on a non-tanking champ. Now I can play Veigar in a whole new way...
Veigar, The Indomitable
Before, whenever I played as Veigar, I was the definition of a glass cannon. Now, I can get into a team fight without having to stand at it's edge. First thing first though, the item build I use now
- Sorcerer's boots
- Tear of the Goddess > Archangle's Staff > Seraph's Embrace
- Glacial Shroud > Frozen Heart
- Catalyst > Rod of Ages
- Abyssal Scepter
- Rabbdon's Deathcap OR Force of Nature
The last item slot is usually determined by whether or not I'm managing kills during the match or if I'm taking on a lot of magic damage (AP damage). If you're getting a lot of kills and if you think you can get away with it, build a Mejai's Soulstealer and replace Abyssal Scepter. You could replace Rod of Ages and keep that magic weakness aura, but you'd lose out on the mana boost and the AP boost from having more mana, so getting rid of the Scepter would be better overall.
Now before all the updates, I would max out Dark Matter and level out Baleful Strike and put a point into Event Horizon after every two into Baleful Strike after maxing out Dark Matter. With this build I'd recommend basically the same thing, except swap out Baleful Strike with Event Horizon. That stun is going to be more useful than the passive, permanent AP boost from Baleful Strike, and since you aren't rushing for an immediate AP boost with this build the stun will help you land more Dark Matters (which is the main, non-ult, source of your damage).
Now you could go for a support role with this kind of build (especially if you choose Force of Nature over Rabbadon's) but you'd be more or less a kill assist and wouldn't really be able to keep team mates alive while Event Horizon is on cooldown. Really the main point of this build is to have enough mana to outlast other AP champs and to bring a massive amount of AP in late game team fights.
Don't get to heroic in the early game unless you can be absolutely sure that your team will help you get away. If you play it relatively safe, while getting a little reckless for kill secures, you should be an unstoppable force late game.