Not Updated For Current Season
Ok, this is just a guide for Karthus. Most people play Karthus as a "glass cannon" so to speak. Personally, I find this not nearly as good as a bulkier version. With this build you can stay alive during teamfights, rather than run in and just get destroyed.
This build will allow you to use your AoE to it's full power and do maximum damage while giving you the survivability you need. It's much more cost effective to stay alive after the teamfight rather than die, like most Karthus.
These are pretty standard caster runes, feel free to use your preference, but these are mine.
Personally, I choose these for the magic penetration that helps ever so slightly. For yellows I go with mana regen so I can keep that main damage source( Defile ) up for a longer time period.
For glyphs I go with Cooldown Reduction so that Lay Waste is even more spammable.
For masteries I go 9/0/21. This is so I can get the mana regen in utility and the magic pen in Offense. I chose the death timer reduction because it goes nicely with his passive. The mana regen is crucial for your Defile as that is your main damage source for teamfights. The magic pen in offense helps amazingly with tank-busting and getting those last hits with your ultimate.
I start with a mana crystal and potions. Then I go into sorc boots. I turn the mana crystal into a tear of the goddess to help with mana regen. Now, I work on getting my health. To start I get a rod of ages ( you want to get this early) then I go into a Rylais Crystal Scepter. Rylais goes well with Karthus' Defile as it keeps the enemy inside of his AoE.
You will want to get rabadons for obvious reasons. Next you should build your archangels. To get your armors up you want Abyssal Scepter and Zhonyas ( build these BEFORE your archangels if you are having trouble with survivability )
I try and max out my Q and E as much as possible while keeping one in W for either escaping, or catching up to an enemy. Obviously you want to get your ultimate whenever possible.
With this setup you should be doing maximum damage with all of your abilities. I choose only one in W because having later ranks isn't that much more important, when compared to your Q or E.
Never use your ultimate to kill steal ( Unless you're playing in a premade..then it's kind of funny sometimes). Killstealing is very poor manner when it comes to a team game like LoL. You want to spread the kills out evenly with all of your carries.
Always use your W to help your teammates escape from danger, and to help the disorientation of teamfights.
Do not overlook your Q as it is amazingly powerful, in a 1v1 situation. This gives karthus the upper-hand if he is fed.
You may be built like a tank, but always let your tank engage and then you swiftly come in with your spells.
Pros / Cons
Has a very high damage output
His (Overpowered?) Passive.
Farms like a champ
Causes rage among your enemies.
Requires a pretty good team, for cooperation.
Can be completely useless if not played correctly ( But this goes for most carries)
Banshees veil can shut down his ultimate
This is my first ever guide, so it may not be the best. However, this build has always proved to work for me. I hope you enjoy my way of playing Karthus.