GENERAL: This Karthus build is for mid and is what I use. Generally I go 8-20 / 2-10 / x depending on how the game unfolds and how long it is etc. For the most part I just use lay waste to farm minions, although if you have good aim with it it can be a good harrass spell, best at 4+ish cause with wall of pain's movement speed decrease its a lot easier to hit someone, even if they know its coming and try to dodge.

Late game its fairly easy to kill just about anyone by wall of pain + defile active and stand next to them while using lay waste.

With your ult at level 6 you can pretty much kill anyone at 150-250 hp, depends on their magic resist, and if they have any spells that reduce damage. To be on the safe side take your ult total damage and take off 50-75.

Around lv10-11 you are going to want to have golem buff all the time, not only does it increase mana regen by like, 150per5 when you are low (chalice ftw), but it also reduces the cooldown of your ult by about 30 seconds.
Note: when you have 20 soulstealer stacks and golem buff your ult can only be reduced to 72 seconds, which is still amazing as it can hit 800+ late game.

SPELLS: Because defiling someone is a rather easy way to get a kill, I prefer teleport and flash as my combination.
Ignite is a possible choice if you keep having people get away at low health.
Exhaust is the other main possibility, a slower person is a LOT easier to hit with lay waste.
Clarity can be good if you are having mana trouble all the time.
Ghost can be taken over flash, but personally I prefer flash.

ITEMS: Healing potion and Tome, or 2x healing 1x mana and regular boots to start off.
Get soulstealer immediately, then get sorcerer shoes and then chalice, by the time you are getting chalice you should be 10-11 ish and so go and get golem buff, thisll let you spam lay waste.
Next hextech revolver, since mana wont be much of an issue anymore you are gonna need health.
After hex get zhonyas, ability power and SURVIVABILITY, very important.
Next up is will of the ancients for even more healing.
And now for Void staff, if the enemy team doesnt have much magic resist you can skip this till after lichs bane. (It's easy to tell if you need more magic pen when your ult hits no where near what it should, with the exception of damage reducing abilities that some heroes have.)
And now that you have lots of gold get lichs bane by selling chalice, keeping blue buff all the time should ensure almost no mana problems and now every auto attack you do will also do like 300-600 damage.
If the game continues you drag on and you have 2k after buying your wishlist, sell your sorcerer shoes and buy elixirs, then buy back your shoes, the cooldown reduction and AP is a nobrainer, agility helps with attack speed / crit which is great with lichs bane, and the extra health is gonna help you stay alive longer.