Just putting this on Mobafire so I won't forget it.

Try to get middle lane.

The way you play is largely dependent upon how much AP you have and who you're playing against. The first few levels should be all about last hitting minions with Baleful Strike and if that's on cooldown then obviously try to last hit with normal attack.

Often you'll be laning middle against another AP heavy character such as Karthus, etc. Wait until level 6, make sure to have Clarity up if you've spent a lot of mana on Baleful Strike last hits. For someone like Karthus, skill sequence against him should be Event Horizon, then immediately (and I do mean immediately), Dark Matter, Baleful Strike, Primordial Burst. Those last 3 abilities should all land at the same time. If they don't then you're not sequencing fast enough. Don't be afraid to Flash to close the distance to your target, but I try to save that for getting away if things go south or if the enemy has a flash of their own and is at very low health. Unless the enemy is playing extremely defensively you shouldn't need to flash to snag them with Event Horizon.

I get Rylai's Crystal Scepter for the health.

Note on the last item: You need to pick it based on the game. I think the best option is Void Staff to bypass magic resist. Abyssal Mask is also a good choice because a lot of the damage against you is going to be magic based, it gives more AP, and obviously the lowered magic resist of nearby enemies helps your entire team. If you're not having issues with health and want more damage, get a Deathfire Grasp. If you're playing against Karthus or if you're getting targeted by enemy AD (Trynd, etc.) get a Zhonya's Hourglass.

Don't forget, if you have your 6th item and have plenty of money, sell your Doran's Ring and pick up an Archangel's Staff for the mana regen. Then buy all the elixirs.

I don't recommend exchanging the straight AP for AP/per level runes because you can get a pretty good bit of kills early on, plus the straight AP runes help with last hitting minions with your Baleful Strike. Just my opinion.

The Greed in Utility doesn't add up to much, but I rarely go for monster buffs with Veigar. If you go for blue buff a lot, then go ahead and grab Utility Mastery .

Dark Matter is your damage dealer, but I still recommend getting the 2nd Baleful Strike at level 3 to help with last hits. The Event Horizon at level 2 is just for preventing early ganks.

Once you get Rank 4 of Dark Matter you can drop it on the caster mobs to one shot them and get tons of gold. If you don't do this you're not going to build up a gold advantage. Note that this is also very costly in terms of mana, which is why I pick up Clarity instead of Ghost or some other summoner spell.