Pros:
-Loads of true damage.
-Surprisingly good laning phase if played aggressive.
-HUGE power spike at level 6, insane kill potential unlocked.
-Good kiting (further enhanced if you build Rylai's).
-Good peel.
-Lots of AOE damage which makes for good teamfight potential and great waveclear.
-Good zoning ability.
-Very long range.
-Relatively low mana costs for an AP mage.

Cons:
-Immobile.
-Prone to ganks.
-No sustain.
-Mediocre scaling on his basic abilities.
-Skillshot reliant, especially his Q.
-High priority target, especially in teamfights. Requires good positioning to survive.
-Hard countered by assassins and champs that can gap close on Vel'Koz and burst him.

Q

Your main poking tool in lane. Use this to snipe people and apply stacks. The slow also helps you kite as well as land your W-E combo. Fire Plasma Fission sideways and take advantage of the 90 degree split to hit enemies behind minions or Wind Wall. Since this abilities restores 20 mana per unit killed, you can actually restore mana with this ability if you kill 3 units with Plasma Fission!

Max first for damage after taking 3 points in Void Rift.

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W

Your main waveclear and combo damage ability. Combo W into E to deal damage and proc Organic Deconstruction. Use this ability to waveclear, poke, or apply passive stacks on enemies in lane. Use Void Rift twice to clear an entire wave.

Take 3 points in this first, because the waveclear and damage you get from this ability is really helpful in lane.

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E

Your CC ability. Use this ability for kiting, peeling, and escaping ganks. This ability also combos well with Void Rift. Tectonic Disruption also briefly reveals the area that you target, so you can use it to check bushes, objectives, etc. BE CAREFUL NOT TO SPAM THIS ABILITY, as it has a high cooldown. Good players will bait out your Tectonic Disruption or wait for it to be on cooldown before going all-in.

Max this last because it is used mostly for its CC rather than its damage. You'll be better off maxing Plasma Fission or Void Rift before this.

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R

Vel'Koz's renowned death laser. Life Form Disintegration Ray has a very long range and does an INSANE amount of true damage if your enemies are researched. Generally use this ability to kill or force enemy champions to back/blow summs in teamfights. You should try to get some of your opponents researched first before ulting to maximize your damage. Also MAKE SURE you're safe before you ult, as it makes you a sitting duck. Track enemy cooldowns in fights, and only use ult when you're sure that you won't die while ulting. Use Life Form Disintegration Ray to shred bruisers and tanks in fights. This ability is also a very good zoning tool. You can use it to zone the enemy's back line from the fight.

Take a point in this ability whenever you can (Lvls 6, 11, and 16) because it is your strongest ability and is extremely useful in teamfights as well as securing kills in lane.

Hands down the best rune for Vel'Koz. This rune synergizes very well with Vel'Koz's kit because it is almost guaranteed to hit if it's procced by Plasma Fission. Arcane Comet adds to Vel'Koz's already obnoxious poke, which makes Vel'Koz really scary in lane. The only other alternatives to this rune would be Summon: Aery and Electrocute, but neither of them are really as good as Arcane Comet.

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OR

[ultimate hat]] is good on Vel'Koz because it helps reduce his ult's cooldown. This allows him to have his ult ready more in teamfights, especially in the late game. [ultimate hat] also lowers Life Form Disintegration Ray's cooldown in lane, which in turn, grants Vel'Koz more kill potential post 6.

You can also take Nullifying Orb for more survivability against AP. I usually take this rune against AP mids with a lot of kill potential like [Fizz], [leblanc], and [brand].

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OR

As for tier 2 runes, both Absolute Focus or Transcendence are good.

Absolute Focus is good for poking as well as teamfighting. Take this rune against enemies who have weak laning phases so you can poke them out of lane. The extra AP is also helpful in teamfights if you can stay safe and not take damage, as the AP will greatly improve your ult's damage. Don't take Absolute Focus against poke laners such as Xerath, because you'll just get poked down below 70% health, rendering Absolute Focus useless.

Transcendence is good for spamming abilities in lane and in teamfights. I generally take Transcendence if you want the cooldown reduction it provides. Also synergizes quite well with Cosmic Insight and the CDR provided by Luden's Tempest.

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OR

Rather straight-forward, take Scorch if you want the early/mid game extra poke damage that it provides, or take Gathering Storm if you want the late gaming scaling AP it provides.

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Magical Footwear works well on Vel'Koz because it provides you with a free pair of boots at 10 minutes, which will help you dodge skillshots more. The free pair of boots will also help you save gold when building into your Sorcerer's Shoes.

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Cosmic Insight is a great rune on Vel'Koz because it gives you CDR that doesn't count as part of the 40% CDR cap. CDR is great on Vel'Koz because it allows him to spam his abilites more. Cosmic Insight also synergizes very well with Transcendence should you take it.