Welcome reader. Since you are here, I can only assume that you are here to kick *** and chew bubblegum. Karthus is your standard AP nuke. He can dish out insane amounts of damage whether he is with the team or not. I've been playing Karthus for quite some time, and have found his abilities to be extremely useful in almost every situation. This guide is will be centered around high sustainable AP damage. The idea is to make full use of Karthus' abilities.

Lay waste is your Q, and is a spammable nuke. It has good base damage and scales .35 with your AP, but if it hits only 1 target, its damage is doubled. This means that when harassing, you should aim for only one target with it. Max this first.
Defile your E. This is an AoE Toggle with roughly the range of your auto attack. This is used for farming late game, to discourage enemies from getting too close, and to supplement your lay waste damage. Only use it for a short time, as it drains mana extremely quickly. It scales at .25, and is your best friend in team fights. Max this 2nd
Wall of Pain is your primary chasing, and running tool. It is also a powerful spell in team fight as it slows by 80%, and debuffs armor and magic resistance by 35 each. Max this skill last. You only need one point early to assist with kills.
Requiem is your ultimate. Anyone who has played against Karthus knows how good it can be. It has high base, and .7 scaling. It auto targets all enemy champions. This is your finisher or initiator. Early game, enemies with base MR will take about 200 damage from it. Remember that number. When it comes to using this skill, initiate only if they are using a low number of banshee veils. Otherwise, wait until you are dead or have escaped the teamfight. DO NOT use it just to KS. Your team needs kills as much as you do. Only use it to kill someone who has escaped beyond your team's range.
Death Defied is your passive. It gives you 8 seconds after death to cast spells, and reduces your death time by 20%. This passive should rarely come into play until late game. You can't be interupted while dead, so it is a great time to use your ult.

Pros:
Great damage, sustained and burst
Strong 1v1
Strong team game
High base mana
Good speed for caster
Great farming and killing potential

Cons:
High Cooldown Ult
Squishy
No hard CC

-Don't be afraid to die, but know when to stay alive
-Suiciding in a teamfight is worth it if your team can keep them next to you. Your passive is useless if there is no one for you to hit.
-Don't KS, you will get fed enough
-Communicate with your teammates. They need to know when your ult is up, and you need to know when you can get a kill with it. Make sure they know the damage you do.
-Gank any chance you get
-If the enemy has a jungler, go to their blue and gank it, or steal it. Karthus with level 1 blue buff destroys lane
-Get blue buff whenever you can. The mana regen and CDR are amazing
-Early game, try to last in lane until you have 950 gold. When you have that, go get a tear and boots. If you have to go back before you have 1600, just buy a blasting wand. Against early MR, get sorc shoes
-You can solo blue buff once you have a blasting wand or rod
-You can take dragon solo at around level 14
-If a your teammates are running away, go for the intercept. Wall of Pain is a great tide turner, and can not only help you escape ganks, but help allies as well.

Your runes are going to be magic pen marks and quints. While magic pen marks are a given, there is variation in the quint choice for many players. I am of the belief that 15 AP doesn't do much with Karthus' poor scaling. Magic pen is still good early game, and makes a much bigger difference late game. The seals are used to assist in laning. They have little value late game, but they keep you in lane longer for more harass. I like cdr per level glyphs. They don't do much for you early game, but they give you late game cdr of around 8%. This is important solely because of your ult. The sooner it's ult, the sooner you can pick off a lucky opponent. Masteries are the standard 9,0,21 for casters. Spec flash and port because you use them. Good hands synergizes with your passive. Meditation helps keep you in lane, and utility mastery helps you get the most out of your blue buff. Quickness gives karthus a much needed speed boost.

There are many item builds that are designed to keep Karthus alive longer. I will not focus on that. I will focus on getting your damage up. While you may work survivability into your build, always rush a deathcap. Many guides discourage the use of Archangels. I use them primarily for the mana regen. If you want to Archangel stack to 4 of them with sorc shoes and deathcap, I'm not stopping you, but don't expect it to work against teams of less than 5 squishies. Throwing in a void staff nullifies resistance amazingly, especially when combined with wall and your 34 magic pen. Hourglass is optional, and I have been favoring it lately, but you can also use just about anything you want. If you are having trouble with AP, an abyssal may be better. Crystal Scepter is also good if you want some survivability, although I'd recommend an RoA over it because the wall is all the slow you need. A 3rd Archangel is also viable. Will of the Ancients is a good pick if you can get a teammate to get one as well. 40% spell vamp is survivability. Of course, you can also get other items as well. This build doesn't work for all situations, and you do need to gauge what you need. You may also choose to use only 1 archangel, but you won't stay in battle for long.

As Karthus, you should be grabbing mid every chance you get. If you can't take a dual lane over solo top. Typically, good karthus lane partners are those with knockups, stuns, and snares. In addition, any champion who rushes a Mana Manipulator will allow you to harass endlessly. During Laning Phase, you must be focused on last hitting minions, as well as harassing enemy champions with lay waste. In dual lanes, it is important to be coordinated with your partner to keep feed off your lane. This guide will focus primarily on solo mid. When laning against your opponent, it is important to know the character they play. I would recommend trying other common mids just to know what they are week against.
The first thing I can say is that even with the mana regen you start with, you NEED to be smart about your harass. You need to keep track of minion health to snipe last hits while dropping lay waste under or around the enemy's feet. If the enemy tries to zone you, punish them. Drop lay waste in front of them, and let them run into it. At a certain point, you may get in the situation where the enemy has low health, but stays in lane. If they turret hug, try to drop lay wastes near them, then back off to avoid aggro. If they over extend, drop your wall, flash in, and kill them. Keep track of your mana while you do this. If you are low, you should avoid the use of defile. In some cases, you may end up with a champion that can easily shut you down in mid. Champions such as Katarina or Kassadin are very hard to lane against for a karthus. When it comes to heavy burst champions, you need to play defensively against them. When it comes to winning these lanes, you have to abuse their mistakes. It is important in any situation, however, that you maintain distance. If the enemy mid goes back or dies, it's time to go help out your allies. Go position yourself to gank. As karthus, you can easily take both champions down with ally support, even if the lane is still not pushed either way. Open with wall, and start attacking the lowest one. If you have your ult, use it to secure escapees. During this part, avoid deaths as much as possible. If you must kill yourself, get your team 2 kills out of it. Still, try to avoid killing yourself. While you are building glass cannon, you don't want to start jumping to your death immediately. Stay alive.

Once you get your Deathcap, Sorc shoes, Archangel core, you should have nearly 400 AP. This is where you will get fed. Kill anyone you can. Again, don't KS, but seize chances to kill opponents. If you are near a tower, push it, but don't get killed. This a the time where a lot of enemies may start to stack MR. This is the point at which you check your enemy's items. Do you see banshee veil, hexdrinker, FoN, Negatron, quicksilver, or abyssal? Get a void staff. If you don't rush out another item.

At this point, you should be near a full build. If you have extra gold, get red and blue elixers, as well as oracles. In the event of a team fight, try to come in for a flank when it's initiated. Wall, flash in, put defile on, and start lay wasting. If you have hourglass, pop the active before you die. DO NOT hit R until you are dead or out of the fight. You won't finish channeling. Getting your ult canceled can mean the difference between your team getting aced, or their team getting aced. As a high AP sustained damage carry, you can destroy Baron with one tanky player. Your dps alone will kill it quickly, but you need someone to tank it for you. Get blue buff whenever it is available.

If you feel that I left anything out or missed a point, feel free to leave feedback.