A good guide to how to play a champion is made simply by realising the strengths and weaknesses of it and how to play into those.
This is what seperates Veigar from other ap carries:

Pros:
*High ap-gains
*Instant Aoe stun as E-skill with longer then normal cast range
*Works as Anti-ap ap carry
*Reaches new heights of ap with Baleful Strikes which escalates further if fed on kills
*Great autohit and casting animations making it easy to lasthit
*His stun and burst also makes him an ideal ganker and often secures kills when a friendly jungler ganks your lane

Cons:
*Only 3 damaging spells
*Easy to miss with stun, doesnt stop dashes
*Very hungry for mana, all game
*Bad lane sustainability, has no form of heals and average base damage on spells
*Dark matter is a small and slow-to-arrive aoe

So, what do we make of this?

Well unfortunately I have to say, though a faithful servant to Veigar, that if you want an ap carry, you'd be better of with Brand or Malzahar. Veigar doesn't do enough damage over time to carry a team. We can get pretty close due to my build though and he also does work as an extremely good counterpick for the deep ap-carries.

I'd say you can pick Veigar if the enemy picks:
*Orianna
*Malzahar !
*Anivia
*Karthus !
*Brand !
*Cassiopeia !
*Fiddlesticks
*Heimerdinger !
*Katarina !
*Nidalee !
*Sion (Ap)
*Twisted Fate (Ap)
Thats actually quite a lot...

Oh yeah need to show you how to build Veigar as well.

Saw those pros and cons? Lets take away the cons shall we?

We need:
*Hp
*CDR
*Manareg
*Deathfire Grasp for another spell to scale with your awesome ap (since we only have three(its like a second ulti!))
*To practice stunning, a lot
*Easier laning
What do we do? We buy a Phil. Stone, dfg and look happy, that's right!
There is no way of getting any other healthregen in your build without ruining it.

Runes:
Magic penetration Marks - Standard for an AP-caster
Manareg/lvl Seals - We need loads of manareg, works with Veigars passive
Ap/lvl Glyphs - Also standard, can be replaced by manareg/lvl as they are almsot as good as the seals.
Flat health Quintessences - Flat health is always good and unlike other casters, you cant go for RoA

Masteries:
9 - For that extra % magic penetration. Works wonders with Void Staff at endgame.
0 - Theres nothing we need here.
21 - Manareg, movement speed, cdr and more flash/ghost

Items and playstyle progression:
Keep in mind, that builds should vary depending on your enemies and how well you're doing.

Get your Meki Pendant and place yourself mid. Brace yourself for some painful laning. It's time to play passive-defensive. Try to farm as best as you can, don't harrass, you will never do it effectively at this level. However, if you find yourself not being to empty your manapool because of severe aggression from your opponent, throw a stun, dark matter and baleful on the opponent. It won't be mana effective, but its safe and it motivates your jungler to gank the lane. Get some lasthits when they're outta range, just used an ability or have to lasthit themselves. You'll always find small windows of opportunity to get those lasthits. They'll outfarm you for now but, thats just the way it is (well in ranked anyway).

Keep on laning til you're out of potions and a bit low on health, play it safe and blue pill. Buy a Phil. Stone as soon as possible to make your laning easier on you, it also makes your endgame stronger by providing more max gold. You're building DFG but dont get Fiendish Codex first, it doesn't give additional manaregen. Get your Kage's Lucky Pick, a pair of boots and some health potions. The enemy ap carry will most likely be going for RoA as ap carries has an incredible dmg output midgame and only lacks in defensive stats, it also helps laning and becomes even better lategame. But you can't use that, it doesn't give nearly enough manareg.

Some time have passed and the enemy goes back to finish up RoA, this is where this build gets strange. But the philosophers stone, kages lucky pick, fiendish codex and kindlegem should amount to about what a RoA costs and it gives:
*Hp (less then RoA)
*CDR
*Manareg
*More ap then roa (atm)
*Gold/5
*Almost a dfg

This + farming should land you at around 100-150 ap. If teamfights still are uncommon (not likely) you should go for Rabadons deathcap while keeping the gold/5 items so you can get it faster or if teamfights are common, you go straight for DFG and Shurelya's reverie, then you get rabadons deathcap.

When dfg and Deathcap is up, the enemy is forced to stack mr. If they do buy mr, buy a void staff. If they continue buying offensive items, buy yourself a defensive item and THEN a void staff. I recommend GA against heavy melee AD teams and Banshees Veil against everything else. That's a complete build!

Summoner spells

Flash: is the best summmoner spell ingame period, endless uses.
Ignite: Many plays with this, I don't. It's range and the fact you ahve to click one more button messes with my play.
Teleport: So good, its a free global ult! Turn ganks into wins or push back lanes or get back in lane fast after failure.
Ghost: no thanks, not good enough.
Exhaust: if noone else is willing to take it... maybe.

Skill sequence

1. Max ulti
2. Max baleful
3. Get stun
4. Max dark matter
5. Max stun
Highest valued guide on mobafire says to get some baleful and then max dark matter, this is wrong because; base values are almost the same and balefuls cd is much lower, you will not lose any dmg by going baleful first, you will gain more actually (because the ap gains stay the same). You will also be able to spam baleful more often, getting more ap. The only place for leveling dark matter first is if your opponent is pushing very hard... maybe.

Usage of spells

Baleful strike: spam it at mobs, get ap, win games. Makes a decent poke in a 1v1 lane.
Dark Matter: Used to push back lanes, kill neutral mobs or as part of your harrass/kill combo.
Event Horizon: Aoe stun, used to escape enemy or to set up your combo.
Primordial Burst: High base dmg, high ap gains ulti which gains extra dmg based on enemy ap.

Combo:
Event Horizon (stun) --> Dark Matter --> Baleful Strike (harrass)
Baleful Strike --> Event Horizon (stun) --> Dark Matter --> (DFG)--> Primordial Burst --> Baleful Strike (kill)

Blue buff: all ap carries loves that ****, however Veigar is not seen as a fully worthy ap caster and your team might just take it instead cuz they are *******s, that's one of the reasons this build doesnt focus around getting it as standard. If you do get it, farm and harrass like a boss!

Recommended team setup (with veigar): I like playing with an ad-tanky heavy team so that the opponent feel reluctant to buy mr. That things go better then might be an illusion brought by me getting more kills, but thats been my experience. Don't think that double ap is recommended because the enemy doesn't even ahve to think about what to buy then.

Veigar as support (?!): Well, it works pretty good with his added dmg and ability to stun both enemies at once, however he does not heal (etc) and is subpar. But if you make it to endgame, it might be a great investment having a support who scares away ap carries. Haven't really tried it in high elo play, don't think I will either.

Well, the rest you have to learn by playing the game people. The only thing I can tell you more is just a general guide of how to function as a team(!)player within LoL, but thats a pain in the ***! Please do tell me of your experiences with Veigar or if you found some huge theoretical flaw in my build (there is no practical one MOAHAHHAHAHAHAHHAHAHAHAHAAAA)! Cya!