I've been a Karthas player for a while now i've found that this build is pretty much unbeatable late game and also has a good early game presence. Explanations for my decisions will be posted in relevant chapters.

These are fairly normal runes for a mage. Flat AP quints and Scaling AP Glyphs provide a good early and late game. The mana regen seals let you last hit and harass without ever having to care about your mana pool. The magic penetration reds increase your damage throughout the whole game.
Other possible runes include Flat Ap glyphs for increased early damage, Scaling Ap seals if you don't think you will need mana or your team is going to give you blue buff and spell vamp quints if you want more lane sustain and late game survivability.
I personally think that my rune set up is superior but it all depends on your playing style.

Standard burst mage masteries going 21-0-9 taking all the AP related masteries in the offence tree and anything that increases mana regen or mana in the utility tree. Take neutral buff duration increase so that you can keep blue buff for as long as possible.
If you think that the build doesn't provide enough survivability you can go 21-9-0 but this decreases your late game damage output significantly and hurts your building of tear.

I choose items almost entirely based on damage output with a secondary focus on survival. The main reason for this is karthas doesn't need to survive to do damage. Building magic resist and armour is a waste of money. You want survivability only when it also significantly enhances your damage. With this build you end up with roughly 1050-1150 AP as well as a decent amount of survivability and unlimited mana.
There are really only 2 other items to consider on Karthas. These are Will of the Ancients and Void staff. I usually skip out on will because i can almost always convince an ally to get one for me. This means i only rarely take will, maybe if i am doing poorly.
Void staff is a situational item. I usually find it is not worth getting because most carries and mages don't bother getting magic res and those are your main targets. Since Maw has come out it has been quite commonly bought against me so this may effect whether or not i actually need to buy a void staff. I dont have enough information on when void staff becomes better then an archangels (which is probably what you would swap it out for). This basically means that i only take void staff when the enemy team consists of mostly tanky bruisers and off-tanks.

Fairly standard skill sequence. Requiem < Lay Waste < Defile < Wall of Pain. Max your ulti and lay waste as soon as possible taking one point in wall of pain early to help escape/chase/gank. Use lay waste to last hit minions from range being careful to only hit one creep if possible. Feel free to spam on enemy as often as possible.

With Karthas i use fairly standard summoner spells in flash and ignite, Flash helps you get out of and into fights (ill explain later) and ignite helps you secure kills. Teleport, Heal and exhaust are all also workable spells but i find them to be passable at best. Teleport is not that useful when you can just ulti for kills/assists, heal is situational and many mids take ignite potentially halving its effect. Exhaust is useful only if you find yourself facing an AD mid particularly talon.

This is the main area in which my Karthas differs from many others. I usually only play Karthas when i have at least a few players i trust, but any good team works. Until level 6 play it safe, use lay waste to kill creeps making sure that you only hit one creep otherwise it is much harder to last blow and you push. In addition whenever no creeps are low try to hit your opponent. Karthas with my build, runes and masteries has a nearly unlimited mana pool so try to harass your opponent whenever possible. When going for a kill use wall of pain to slow the enemy then turn on defile and try to predict the opponents movement with lay waste. Do not chase into tower for any reason it is not worth it. If your opponent is on low hp but is escaping this is when you use your ultimate. two important things to remember at this stage of the game are, If your opponent jumps next to you DO NOT RUN, unless they clearly hold a massive advantage. Karthas does more damage, it is that simple. In a 1v1 fight almost no other mage can deal as much damage as karthas and in addition to this even if you die you can often get a kill. Since there is no penalty for dying in LoL (apart from first blood) dying for a kill is worth it for your money hungry late game build. The other important thing to remember is to tell your teammates to CALL FOR YOUR ULTI. I cannot emphasize how important this is. More times then i care to remember have i got kills when an ally calls for an ulti. Always tell your allies when your ulti comes back up, and make sure you retreat to a position where your ulti cant be cancelled to use it. THE ONLY TIME you should be using your ulti when you are not dead is when you are helping to kill another lane and you are 100% safe and when you are using it to kill an enemy champion who is fleeing from you and is WAY out of CC range.

Important Note: For the love of god, do not intentionally steal kills from the AD carry. Any other character is fair game.

Once you reach late game this is the part where my Karthas play style differs from most others. i play karthas as an INITIATOR. Yes you heard me, I initiate as Karthas. The best way to win teamfights as Karthas is to flash into the enemy team with defile on lay down a wall of pain and take as much damage as you can before activating zhonyas. Not only do the enemies ALWAYS immediately go for you, they will often all run directly at you if you leap in thinking you are an idiot and an easy target. Karthas's defile does insane damage litteraly killing some squishies in 5 seconds. As they focus you down you are doing a large amount of damage and keeping the heat of off your other damage dealers and healers. As Karthas you WANT your enemies to focus you. Zhonyas helps uncoordinate them as they wait until it runs out to cast spells on you wasting time and damage. Even if you do die you CAN CONTINUE FIGHTING, meaning that even if they do focus you down and kill you you can still fight. THIS IS A HUGE ADVANTAGE GIVING YOU SEVEN EXTRA SECONDS OF DOING DAMAGE AFTER DEATH. This is the reason why Karthas should lead in team fights. It makes the enemies use to many resources to kill an enemy who keeps fighting. Make sure you use your ulti no more then 2 seconds after dying so as not to lose it. This play style isn't for people who want to look good on the score board as often it will result in a trade of of you aceing your opponent for you as the only death. Having said that my average score still ends up around 20-8-20.

Pros

- Good early Harass and farm
- Difficult to Harass
- Massive late game Damage
- Unrivalled team fight presence

Cons

- No real crowd control
- Susceptible to high mobility enemies
- Banshee's veil and Maw of Malmortius hurt alot

Best allies

Shen, Soraka, Kayle, Galio, Amumu, Basically anyone who can keep you alive longer, help you get into a fight or help to keep the fight near you. Any of these heroes greatly increase your effectiveness. If your allies ever ask what hero they should play suggest any of these four.

Best enemies

Any AD carry, Any Mage reliant on skill shots that are blocked by creeps and any Squishees that have to close into melle to deal damage. Good examples include Caitlyn, Brand, Akali and Katarina but most mids except the ones being mentioned later should be easily beatable. Anivia, Fiddles and swain are all easy wins as long as you play smart.

Worst Allies

Orianna, Janna, Garen, Xin. it is not that these heroes do not work well with Karthas but that they will often really screw you over. Janna and xin will often knock people away from you when you are dead which is the WORST POSSIBLE THING that can happen. Orriana has a similar problem occasionally and Garen will regularly waste his ulti on something you were going to guaranteed kill anyway.

Worst enemies

Master Yi, Ahri, Galio, Kassadin, Fizz, GOOD talon, Soraka and Shen. These heroes are the only real counters to Karthas that you will see mid (Except shen and soraka who dont need to be mid to piss you of). Master Yi is unkillable for karthas unless your jungler ganks. His heal nullifies your ulti completely and prevents harrasing and you have no way to stop it channeling. His ulti also prevents you from slowing him your only form of CC. Ahri is highly mobile, hard to harass and deals large damage. Good ahri will usually beat you unlees you have a good jungler. Galio is every AP's worst nightmare, Hes tank, Hes got a crazy Magic res + Magic res boost from his shield, Heal and Good burst. Kassadin is highly mobile has a long silence and good burst as well as a passive which soaks a good amount of your damage output.
Fizz is your worst enemy. Capable of harrasing you then getting away EVERYTIME, Dodging your spells and killing you very quickly Fizz is the hardest ooponenet for Karthas. Talon is very good if he plays smart. Whenever Talon comes towards you put on defile, good talons will use this to drain your mana but with out it he burst you then walks away while you are silence. Soraka and Shen can protect anyone anywhere on the map from you and both have ways to prevent you from fighting (Silence, Taunt).

In conlusion Karthas is one of the strongest mids in the game.
He has excellent early harass and farm, very good late damage, decent utility.
Make sure you get blue buff whenever possible, always using lay waste as you move towards your lane so that your tear charges faster.
I hope this guide helps.