Veigar’s will damage the first two targets it hits, so try to stay behind at least two minions when trading with him.
Veigar can't farm his wave and poke you at the same time. Also he has an really hard time stack AP (from his Q) when he's pushed under the turret. Push the wave early if it benefits you.
Veigar will stack 1 AP every time he hits a champion with an spell or kills an enemy (minion or champion) with his . Try not to get hit by too many spells to hold him down a little longer.
Veigar is limited to what he can do if there is no stunned target. Play around his Cage cooldown as this spell can lockdown an area in a fight for 3 seconds. QSS or Cleanse can be useful in this situation.
Veigar has no mobility spell and once his Cage is placed- he has no way of stopping or escaping an engage from the enemy. Try to use this time window to find an good engange.
Veigar's biggest DPS output is his Dark Matter which has an 1.25 sec delay before hitting the ground. Dodge this spell under all circumstances, as this can easily deal more than 1000 damage later on.
Veigar's hardest level spike will be his level 6 where he can burst you with his which deals damage based on your missing health. Do not overstay your welcome when low.
Veigar will reach a huge damage spike when he gets his Rabadon's Deathcap,as this will increase not only the AP from his items, but also the AP he gets from his stacking Passive.
Veigar has an item build that gives him a lot of health, but even more dangerous is his ability to slow you down with his active items (like Twin Shadows) and then place his cage on top of you. Back away as soon as he starts running towards you as he may be setting up his Jungler.