This gives karthus some ability to push towers. Thats about it. It is weaker in terms of spell damage and has no defense except for speed. This means you should run away most all the time.

Get tear. Max this asap. Spam Q on tear cooldown, even if your just wandering around.

Once tear is maxed and you have seraph. Get another one. Turn this into manamune asap.

Once you have Muramana, your excessive amount of mana gives you some quality AD, even without turning it into Manamune. Nashors gives you the attack speed to utilize this AD boost.

This can work in any lane, but it is gold hungry so a solo lane or jungle works well.

Jungle is a great source of gold for Karthus. Make sure to see that Q hits only one of the monsters when jungling, this important for all but the bird. It gives 2x damage.

Runeglaive is amazing because it gives 8% mana back after each spell cast, so when you have 3000 mana, this is a hefty amount of mana.

To jungle replace Gauntlet with Runeglaive. You lack the spellblade slow, but you will be able to get easy money to complete the build.

Swiftness because this lacks any defense for the most part. This means run away. Alacrity and W will keep you safe for the most part. You can also use Q and blast enemies that are chasing you and stay out of range if they don't have any speed boost abilities and they dont have swift boots.

Seraph/Raba - Normal AP stuff for karthus

Gauntlet- This gives you some armor, AP, and CDR, which are all nice but more importantly mana. The spellblade is nice when wall is off cooldown, because this doesn't have rylai.

Mana gives you both AD and AP in this build. Stay Classy.
Hybrid Penetration because why not both.

Q is quality blast.

W is good peeling.

E is mana hungry, but its nice to have even when you don't use it, it refunds mana on kill.

R is global damage, it is a very small amount of damage without things like void staff and sorc shoes, but it is still good to help turn the favor of a fight that you cannot reach. It has a long cast time, do not use it around enemies. It is easy to use when dead.

Take them as you please, but Q healps you get kills in lane, and even just one point in E and W is effective. Taking points in W is more advisable because E is very mana hungry early on, but putting points in also gives more mana return on kill.

Pros
Can push towers better than an AP class normally can.
Non-build pros of karthus - Global Damage Ultimate, Good Peels and kill secure
Good lane clear (even better with runeglaive)
Amazing amount of mana

Runeglaive Pros
Amazing Mana regen in jungle
Easy money in jungle / Easy heal when low health, just go smite in jungle

Cons

Very fragile and little defense.
Very slow build; it takes 20-30 mins to max out a tear. One cannot get another tear unless one has a maxed out tear and has upgraded it to seraphs, that is when it loses its unique passive.
Less Damage against champs as a build with penetration

Teleport. This will get you to a wave of your minions that has collected some steam and is near a tower.

Flash- This gets you out of trouble... sometimes...

Garrison - This reduces tower damage, which gives you more time to wail on it while your creeps die slower.

Ignite - This is clasic mid lane spell.

Heal- This is health, and since you have 0 bonus health, it amounts to a respectable percent of your total health. The bonus to speed also gives some quality escape for you and any allies that you are with when running from enemies.

Lane Karthus
Mana is good. Keep the slows down.

Jungle Karthus
About the same, just picking up the decent low point jungle mastery.