Veigar is a really, really weird champion at the moment, and it seems like most everyone plays him ever so different. I guess I should give my two cents then, too.
(This is in medium gold - low plat in terms of normal MMR, I don't like to queue for ranked due to reasons undisclosed) My playstyle with Veigar is heavily inspired by a combination between Irelia, Vi, and Viktor, if that makes sense.
In pre-match gear, I use : Masteries : 21/0/9 (Max %AP and armorshred masteries, Movement speed, mana regen) (Often I like to toy with the 1% vampirism utility mastery, but that's another story) Runes : 9x magicpen/hybridpen (in your preferred configuration) 9x armor/health (flat/per level) (I always use (flat) armor so I can tank minion waves a bit longer while I line up Q's) 9x MR/AP (per level, flat MR is an option) (I've seen people run CDR, and I like that option as well, but MR/lvl is just too strong later on in blind pick configurations) 3x MS quints (racecar semibruiser veigar, hear me out) Teleport/Flash
Now, before the flaming, criticising, and whatnot head to a startoff, give me a paragraph to elaborate. Veigar's level 1 is strong, but not all-in strong. Level 2 (and onwards) might have all-in potential with a jungle gank, but a lot of damage will have to come from the jungler, which makes Veigar's missing rune AP slightly irrelevant.
Veigar himself won't have massive kill-pressure (without, and probably even with Ignite) until he has both his ult and about 100-140 AP (we're talking, say, level 6-8). At this time, his Q passive will probably give him 70 AP, a Fiendish Codex will supply another 30, Doran's will give 15, and your masteries will also give some. Plenty. Hit one Q, land E, kill with WQR.
After you can get a lethal drop on any enemy champion in any lane, you can abuse this, Ignite will be an utter waste of precious space, and your AP stacking will surge. If you fail to succeed in getting a lethal drop, this will either mean your damage falls slightly short (which means : wait till the enemy builds more AP, till you stack more AP, till you level your Q/W, or just land one more Q before you go and murder) or the enemy is beyond your grasp regardless.
The reason I use teleport is three-pronged. 1. Veigar is an absolutely marvelous duelist against most champions (I'll give a shortlist of counters further in), as long as his Q deals a noticable fraction of their health, he can kill them. This, combined with his high AP and wavepush makes him a big splitpusher. 2. Global presence on a Veigar is terrifying in its own right, moreso than a Shen ult I daresay. 3. Veigar's early is rough, but if he can quickly back into a Fiendish codex and a mana potion, he can force his opponent to stay in lane at a disadvantage. This advantage can easily be pressed to the point where Veigar becomes a lane bully.
Build path : (again, weird, but explained) The simple idea is this : Once Veigar reaches 600-700 AP, he doesn't have necessity for more power. At this point he can QWR any squishy, and his QW chunks squishies for >50%, and tanks for ~15-30% depending on health and resistances. More AP will increase the chunking characteristic, perhaps even causing 1hko QW's, but will also make Veigar a far easier target.
I believe Veigar has two absolute core items : Rabadon's Deathcap, and Void Staff. Unless the game reaches an incredibly-late-game where his passive alone gives 500AP, Veigar won't reach 6-700 AP any easier than Raba's, and Void is a necessity to counter defense against his pressure.
Next to this, you'll want an Athene's Unholy Grail. Only consider Morello if the enemy team doesn't deal magic damage, and even then consider Frozen Heart. The AP from this item pick is a nice bonus, but the CDR, mana sustain and added protection is what I prioritize for.
Boots : Sorcerer's asap to counter unholy grail and mantle buyers, along with rune MR. Upgrade with HG or Alacrity. In post 30m fullbuilt games, replace with Ionian Boots-Alacrity for maximum teamfight potential. In counterpicked lanes, Mobiboots.
This leaves two spots which most people fill with a pick between Rylai, Luden, Zhonya and GA. I fill this, most of the time, with durability items, like Randuin's (wasted active unless running), Righteous Glory (outright terrifying if used correctly on Veigar), Warmog's, Banshee's Veil, etc. Why? Because Veigar, in the late game, will become a prime target, which means you'll be zoned out of engagement because of assassin pressure in 8/10 games. Stacking over 3.5K HP will stop the cheesy tactic of dropping a Leblanc on you. Stacking over 100 armor and 3K HP will allow you to kill junglers and toplaners before they get you. This kind of magic only happens to full AP veigars if they still have their ult; semibruiser Veigar can kite, stun, outlive and damage for days.
PS : It's late at night while I'm writing this, and my explanation is likely shabby at best. If you're really curious, give it a shot and see if this works for you. I know I really dig it, but it might just be that I'm getting the best out of a subpar build simply out of cheesiness.