Hey there! Veigar's a REALLY fun champion, even if he isn't as mobile and doesn't bring as much utility as other midlaners do. His damage potential is off the charts, and unless he's ENTIRELY shut down in lane and throughout the midgame you can't really stop him from becoming an absolute monster in the late game.
First and foremost, learn your Q damages. If you cast your Q and either the minions get it before the Q connects, which happens more times than I care to think about, or your Q doesn't deal enough damage and a minion gets it, you just lost out on gold, mana, and that 1 AP bonus. Last hitting is absolutely essential for him, both with your AAs and your Qs, because you have no escapes and so pushing your lane is extremely dangerous for you.
Your W gives enormous zoning potential, although it's a bit pricey to use mana-wise during the laning phase. It has a 100% AP ratio on top of the base damage, so it goes without saying it's going to hurt no matter what stage of the game you're in. Don't be afraid to use it to poke at an enemy champion or to clear a weakened row of caster minions after you've got enough AP to make it eat them alive. Enemies can see where it's going to fall before it does, which is usually enough to get them running, and if you learn how to lead it you can control where you want them to go and when. As you're running it's invaluable to get some damage on chasing enemies while you prepare to turn and reign down your unholy Yordle vengeance upon them.
Your E is awesome. An AoE stun, and quite a long one at that, that while it starts off with a meh duration eventually last long enough for you to get your entire combo off on one poor soul while halting an entire team of enemies long enough for your team to come crashing down on them in fights? His other abilities hurt, but this is the ability that will make or break a good Veigar. Get good at setting it up to where an enemy will be standing on top of one of the walls when they rise; I play with everything smartcasted and can land this 9/10 times spot on.
Do I even have to say anything about his ult? Hitting R with Veigar is like shouting "FUCK THIS ONE GUY IN PARTICULAR". Learn it. Love it. Build CDR to abuse it.
Don't be discouraged if you have a bad laning phase. Most Veigars get a bit pushed back during the early levels, and any midlaner worth their salt knows you have to try and deny him farm. Deny them denying your farm with smart use of your W, get stupid good at last hitting, and wait for that magic level 6 to land an E W Ignite Q R kill and then taunt on their body for a little while.
Just don't do it so long the jungler gets you.
But always keep in mind that no matter how janky your early game is, so long as you keep up on your farming when the roaming starts and land team stuns for your allies, you're still going to hit the lategame like a truck.
A tiny, incredibly evil truck.
Happy ulting.