I start out with a dorans ring and health potion. I choose doran's ring over meki pendant because doran's ring gives 5mp per 5 seconds, health and 15 ap, whereas meki pendant just gives 7mp/5. It is only a 2mp/5 which is completely insignificant compared to the health and ap the ring gives.
In addition, we are not going tear-> archangels, nor are we going deathfire's or chalice.

Special mention about chalice: I have tried this build before in replacement of a doran's ring early, but It has hindered my farming capabilities and set back the other items too much to be of real value. Yes given that you have no more mana problems you could farm more, but you are giving up an invaluable early game bonus, which is key to this style of gameplay.

Towards the end of early game, you should have enough money to buy catalyst and boots. This allows to continue to stay in lane for a longer period of time as you level up. I personally do not go back until my health is low and I have enough money for both.

Get Rod of Ages ASAP, you should be around level 9-11. You can start ganking now, as you should have about 125-140 AP, depending how well you farmed early game. Each successful kill is an extra 5 AP.

After that, farming should be a breeze as you have enough AP to one hit the ranged creeps with level 4 dark matter.

The reason I go Rylais is because the health and AP. The 80 AP is a nice bonus, but the real deal is the health. It allows you to stay alive and running, continually dishing out damage in team fights, and more of tank than half your team. Oh. the slow is nice too =]

Deathcap just for the huge-*** AP bonus, 30% extra AP ... ******IC.

Then finish off with lichbane and zhonyas.
Lich bane lets you do a huge damage, but I admit, its hard to master. I find myself forgetting to attack afer each spell as you are just trying to chase them down after you nuke them so hard ;]

Zhonyas lets you survive in team fights and gives you the armor to be even more of a tank. 2 seconds lets you pull off another Q, and puts the focus off of you, which is going to be often because they know how strong you are.

THE REASON I DONT GO MEJAS
You will die. you will LOSE AP. its part of the game, and getting this will just give the other team another reason to focus you. They wont realize the massive AP you have due to your choice of items, but by then it will be too late.

The whole BASIS of this build as said before, is FARMING. You need to take advantage of the 1 AP per creep kill with the nuke. This will allow you to be OP mid game, and just plain scary late. 1 AP doesnt seem much, but think of it this way, flat out ap runes give 25 at level 1, and every wave is 6 creeps. Assuming you can use your nuke 2-3 times every wave to last hit creeps, thats 2-3 AP every 30 seconds. Within 5 minutes of gameplay, you already have surpassed what AP runes can give you, and then more.

Farming with your nuke is underrated when people play veigar, and doing so is essential to dominating early game.

I start out with a dorans ring and health potion. I choose doran's ring over meki pendant because doran's ring gives 5mp per 5 seconds, health and 15 ap, whereas meki pendant just gives 7mp/5. It is only a 2mp/5 which is completely insignificant compared to the health and ap the ring gives.
In addition, we are not going tear-> archangels, nor are we going deathfire's or chalice.

Special mention about chalice: I have tried this build before in replacement of a doran's ring early, but It has hindered my farming capabilities and set back the other items too much to be of real value. Yes given that you have no more mana problems you could farm more, but you are giving up an invaluable early game bonus, which is key to this style of gameplay.

Towards the end of early game, you should have enough money to buy catalyst and boots. This allows to continue to stay in lane for a longer period of time as you level up. I personally do not go back until my health is low and I have enough money for both.

Get Rod of Ages ASAP, you should be around level 9-11. You can start ganking now, as you should have about 125-140 AP, depending how well you farmed early game. Each successful kill is an extra 5 AP.

After that, farming should be a breeze as you have enough AP to one hit the ranged creeps with level 4 dark matter.

The reason I go Rylais is because the health and AP. The 80 AP is a nice bonus, but the real deal is the health. It allows you to stay alive and running, continually dishing out damage in team fights, and more of tank than half your team. Oh. the slow is nice too =]

Deathcap just for the huge-*** AP bonus, 30% extra AP ... ******IC.

Then finish off with lichbane and zhonyas.
Lich bane lets you do a huge damage, but I admit, its hard to master. I find myself forgetting to attack afer each spell as you are just trying to chase them down after you nuke them so hard ;]

Zhonyas lets you survive in team fights and gives you the armor to be even more of a tank. 2 seconds lets you pull off another Q, and puts the focus off of you, which is going to be often because they know how strong you are.

THE REASON I DONT GO MEJAS
You will die. you will LOSE AP. its part of the game, and getting this will just give the other team another reason to focus you. They wont realize the massive AP you have due to your choice of items, but by then it will be too late.

Magic Penetration - Basic to any nuker
Mana Regen - Lets you put out that nuke every time its not on CD
CD Reduction - Same reason as above and for quicker kills
Health - I like going health because the 100 extra health with Dorans ring lets you have plenty of survivability early game.

The reason I don't go AP at all is because these runes allow for survivability and increase your farming capabilities. Flat AP runes give about 25 AP at level 1, which is only 25 creep kills, something that is easily achievable with the CD reduction and mana regen given instead. To get CD reduction for seals is pointless as 9 of them only gives a minimal of roughly 2.7%, where as the mana regen per level gives you 4.5 mana every 5 sec at level 5. With masteries, your passive, and dorans ring, you get over 20-30 mana every 5 seconds at low levels, enough to let you last hit for quite a long time

Max your main nuke ASAP, you want the most damage for easy harass and easier last hitting. And you want the 5 AP per hero kill ASAP as well.

Max your Dark Matter spell next, as at level 4, as long as you are farmed in AP, you can 1 hit the ranged creeps.

Your stun changes team fights, but level 1 is good enough to pull off your combo early game.

Ult whenever possible - this thing is amazing against other nukers, as it does 1 additional damage for every AP they have.

Combo - Nuke-> Stun -> Dark Matter -> Ult -> Nuke

If you have lich bane, attack after first nuke and after the dark matter, If you attack inbetween the stun and the dark matter, you may not be able to pull off the combo completely.

Flash - To Escape, To chase, mind games, nuff said.
Teleport - MAP CONTROL. You see 20 creeps pushing a tower? Teleport and dark matter the whole wave.

Reasons why I dont choose others.
Ignite - You wont be focusing on killing early game, and late game, your burst damage outdoes the significance of ignite.
Clarity - Mana isn't a problem early game due to runes and this build. You can go this, but its useless unless you spend mana unwisely.

The reason I wrote this guide is because I love to play veigar. It lets me surprise anyone I play, and it will be fun for you too.
You can hate, but before you do, try it out =] You wont get the frustrated- he's soo squishy, he dies too much, I cant survive to kill anyone, but rather the: THATS RIGHT FEAR ME attitude in now time.

If you want, you can check out my stats - ign: musicoverlol - fyi, besides the veigar ones, its not pretty.