This is how I personally build Veigar and my strategy with him, so you can see some differences in terms of item build. My idea of Veigar is that he is a 1 shot glass cannon, capable of killing casters or straight ad builds in a 1 shot combo aswell as doing great damage with Baleful Strike in harassment mode, and on occasion if they are glass cannon ap champs, Veigar's ult alone can do the job. I find Veigar is weak during early game and his job is to farm ability power with his q so laneing with someone who has early game advantage like Garen. will be a tremendous help to your farming potential, Veigar is a anti mage class, and personally I find it hard to fight ad ranged in mid lane so avoid this situation Veigar has low hp early on and his baleful strike dosnt really do as much dmg as most ap carry casters until it hits lvl 3 then it is pretty much on par with the others, so until you hit lvl 5 it's all about farming up ap, if you are doing particularly well by the time you hit 5 in your lane you should continue to farm as much as possible because every single point will add up, if you are being harassed by a ap, use your minnions waves to your advantage since your spells wont be on par with them, so kite into your minnions and allow them to initiate then baleful strike them+ the minnion hits should break you guys even. the problem with this build regrettably is his mana will go down pretty fast, the only mana regen with this build for a long time is the mekai pendant, for farming this is perfectly fine, when u hit 0 mana, his passive is most effective, allowing you to continuously cast baleful strike which has a mana cost of 75 at all levels, and by lvl 5, you can harass efficiently with it, it will do enough dmg to send them back to heal and you can continue famring, the problem with this however is if you need to ult or stun a potential gank to get away, you may be **** out of luck, but if you stay map aware and/or if you suspect something you can forgo farming for a short time and have enough mana to stun or ult from the mekai pendant and Veigar's passive, and only cast when you have surplus of mana until you regain sight of the other enemies on the map. on the other hand you don't push well at all in the early game doing this so that may prevent a jungler from trying to gank you, that is presuming your lane isnt pushing too far and in any case you can flash away. by lvl 6 you can harass with ur baleful strike until they are low enough for the 1 shot combo, E, W, Q, R. That's event Horizon, Dark Matter, Baleful Strike, Primordial Burst. this is awesome!, also that changes to E, W, 1, Q, R once you have deathfire grasp, it's a little tricky to get used to but it is pretty much a guaranteed kill, flashing to stun is great because it adds a element of surprise :) - as far as items the abyssal scepter is great tool against other ap casters, if they are ap i'd reccomend getting this first other wise you can get the hourglass first if they are ad it's up to you, if you find you dont need the abyssal sceptor cause they are all ad or you dont need the hourglass cause they are all ap then you can replace with a rylais staff, added hp for survival and the slow helps land your darkmatter time to time, and great amount of AP for a single item slot, another potential item is void staff if most of the enemy team has magic resist it may be neccesary to fit this in to your build. this is my first guide and i know it's not really thorough/complete and maybe a bit sloppy, i wrote what I could in 15 minutes and will continue to add in all likeliness, but the main idea is the item build with very high AP(especially if you farm 100 or 200AP with your q)that will add to your deathfire grasp's active ability, so try it out let me know if you liked it or what you disliked about this particular build, or if you have suggestions to alter something let me know and explain why/reasoning, good luck and have fun. ((p.s. I use flat mana regen seals. I only chose mana regen/lvl in this guide because this build is seriously mana neglegent and I dont have those runes to test so give em a try if you have both even better you can test it and let me know :P, and usually you can get 1-2 kills and recall for mana and items. the flat mana regen may be why i am able to continuously cast baleful strike at lvl 1-5, where as manaregen/lvl seals has less mana regen during this period.))