I don't usually help with leash, let bot or top lane do that. You don't want to fall behind in XP.

You mainly want to focus on farming with Q. Try and hit two of them at once. Also try and last hit one minion and hit the enemy through it.

When vs melee champs try and auto attack them for levels 1 and 2, this does way more poke damage than you expect and you can often get them down to half health this way.

Try and only use Q and auto attack the minions to get them low enough. Don't push the wave. Use W to last hit only if you are being zoned. Using W will cause you to go OOM. If you do go OOM before 1300 gold (which I always do), use minion dematerializer on cannons to help you get that last couple hundred gold before recalling.

Try and last-hit with autos as best as you can. It's hard to do this as a mage because your autos will be very weak, but it is doable and will often get you a higher CS than your opponent.

With Veigar, the farming doesn't really stop after you get lost chapter ... at that point you want to farm for 1500 more gold until you can afford Hextech GLP. This is the item that will let you easily set up cages and kills. It is often worth delaying boots until after you get this item if you don't need them to dodge skillshots.

There are a couple exceptions to the "rush Hextech GLP" rule ... if versus a hard AP matchup, get Null Magic Mantle and Boots after Lost Chapter, and build Banshee's Veil after GLP. If versus a hard AD matchup, get Seeker's Armguard after Lost Chapter and build Zhonya's after GLP.

Veigar has strong pick potential due to his E, so after you get Lost Chapter and have decent waveclear, you should definitely aim to wander into the jungle after each wave. Either just put a ward down, or maybe hover near your jungler ... just don't hug your tower for 15 seconds when there are no minions, that's wasted time.

There are a couple different scenarios.

1. Their midlaner has priority, sees you as prime meat and is trying (successfully or not) to kill you or push you in. In this case you should either continue to try and farm, or ask another laner to swap with you.

2. Their midlaner has priority and is doing the waveclear-and-roam thing. In this case you generally don't want to follow them, but rather spam ping that they are missing, try and set up vision around your lane, and pressure their tower. With his AP stacking Veigar actually does do decent damage to towers, and you certainly can take first tower in many cases as well as keeping your farm and XP up.

4. You have lane priority or you are even. In this case you should decide whether you want to go for a kill or try and roam. Fishing for a kill with Hextech GLP is very straightforward. You get within GLP range, use it to set up a cage, then press the rest of your buttons. Try not to go too crazy for this play, it's the type of thing where it either takes them by surprise and works, or it doesn't. And if it doesn't work (say they dash out of your cage range), you don't want to be way overextended. On the other hand, if you don't feel like trying to kill your enemy laner, you can certainly go for a roam and try elsewhere.

After you have your first item you should be aiming to snowball your income with kills. If you are behind, it's ok to farm for a defensive item like Banshee's Veil or Zhonya's. If you're ahead, you can go straight for Deathcap after Sorc Shoes / GLP. You should aim to get Deathcap as your third item or, in the worst case, fourth.

Stay near your team. Veigar is very vulnerable to assassins like Twitch, Rengar, Khazix, etc and so you cannot split push or go solo to clear a side lane.

Veigar's entire kit revolves around his E and this is certainly true late game as well. You can use it to engage (a flash + E combo on a group of enemies can be really brutal) or to peel for your teammate who's being chased. You are a powerful nuke late game and you should focus on high priority targets, as any carry should. This is generally made a little easier because the enemy carries are usually the most squishy ones as well. In team fights you should not necessarily focus so much on stunning a particular person, as this can be quite difficult, but rather disrupt the fight by throwing the cage into the middle of the fray.

Also, come late game Veigar's Q and W are on quite low cooldown, so he starts to feel a little more like a battlemage and not just strictly burst. But, none of his skills are targeted, so you do have to put some effort into making your casts accurate. It will certainly pay off though. Once you have 1000 AP your W is a legit nuke and you can 100-0 a squishy just with W+Q.

You can do the same thing with Q+R at this point. For example if a Rengar jumps on you, you can simply Q+R them and counter-assassinate ... it's brutal.

In a normal game my AP with Veigar usually caps out at around 1400 with a full damage build. It is possible to get it higher than this but you need to be very diligent about stacking, and resist the urge to W huge waves.

That's all for now ... good luck and have fun ;)