Not Updated For Current Season
So you've got your items planned out its time to start the game. First of all you don't HAVE to do this but I personally ALWAYS solo mid as veigar. ALWAYS. This is becaise as veigar early game you want to farm up Baleful strike as much as possible. When there is only one person on your lane its MUCH easier to last hit. But anyway moving on. So the minions come out and heres what i do. From lvl 1 to lvl 4 all i do is FARM. Farm, Farm, Farm. Wait till a minion is low health then hit it with Baleful strike giving you 1 extra ap. The only thing you do is farm, COMPLETELY ignore the other champ unless theyre attacking you, dont cast on them just creeps. so youve been farming for a while got a bit of ap now fair bit of cash, but have ran out of mana. Even if you are out of mana though DO NOT go back until youre atleast lvl 4. So youre out of mana, lvl 4 its time to go back, but make sure there are NO minions attacking youre turret. Go buy. Now you can harass the other champ when your Baleful Strike is at lvl 4-5. Gank when you see the oportunity. Stun is for starting a fight and escaping.
Combo: E(stun),W(right after E realy quick),R(if possible) and Q(AP)
That's it, you can take other thing than clarity and flash but that's the way I play him
P-S: If they are always targeting you and you die, get guardian angel.