Sometimes it can benefit Karthus to die a strategic death, by rushing into a group of enemies and casting Defile the enemy will focus you long enough for your team to initiate without affecting your chance of winning a team fight, as Karthus can cast his spells even after dying.
His passive allows him to cast spells while dead. Abuse this every time you die and punish any who remain near your corpse, casting Wall of Pain if not already used and endlessly casting Lay Waste. This may actually get you a kill.
When focusing on harassment, try placing Lay Waste where it will hit the champion, but not the minions. This way, it will deal double damage.
Taking 1 level of Wall of Pain gives you a slow allowing you more time to cast Defile on fleeing enemies, as well as deterring attempts of harassment, even at their towers. In later game it is useful in team fights, thanks to its magic resistance debuffs. With this spell, Karthus can actually be a very good ganker, as it slows and allows his teammates to deal more damage.
Karthus'sDefile at higher levels can mow down creeps by standing next to them or walking past them for just 2 seconds. If given free rein or adequate protection from enemy champions, this can turn the tide of a battle very quickly or cement a teams victory.
Ask your allies to help point out when you should use Requiem to get kills in different lanes, or alternatively, alert your teammates when you are about to use it. However, be sure to keep your eye out for action elsewhere on the map; you may be able to help someone when they otherwise wouldn't have time to type for help.
Before using Requiem to kill a retreating enemy, consider the shield/heal/defensive abilities and items of the enemy team; at endgame, it is usually more important to have an ultimate up than to force enemies to heal.
During laning, it may be more beneficial to attack the caster minions in the back and to kill them than to harass an opponent. If you find yourself being outlaned by the enemy, pushing them back and stacking up on your own minions can help control them and keep them away from your tower.
If your team can effectively distract the enemy, Karthus can actually be a very effective pusher, as the sheer speed at which he can consistently clear minion waves allows him to push his team's minions to a tower and destroy it in a short amount of time.
Requiem can be used mid-game before a major offensive is launched to weaken all enemies, making them easier to kill.
Flash and Ghost are useful summoner spells, as Karthus often needs mobility in order to be an asset to the team.
It is possible to jungle using Karthus with the proper runes and/or masteries. To get the blue golem rune, use Lay Waste only on the blue golem (missing the smaller monsters). This way, you will deal double damage to only the golem.
Be careful of when to use Requiem due to your ability power and the enemies magic resistance.
Using Karthus'sDefile will deal damage every second to your opponent after activation. With no cooldown reduction, it has a 0.5 second cooldown on both ends (activating and deactivating), making it deal the same damage as just leaving it toggled on. With any cooldown reduction, however, you can increase your damage by rapidly toggling on and off. At 40% cooldown reduction, you can effectively deal damage once every 0.6 seconds, instead of once every 1 second, for 1.67 times as much damage.
Wall of Pain can be a viable tool to saving an ally champion's life due to its movement speed reduction. Use it to give your allies a chance to get away if they are running from a pursuing champion.
On Dominion, Karthus can work as a potent defender in bottom lane (between the Boneyard and Quarry), thanks to his potent zoning, ability to easily defend his point until reinforcements can arrive via his passive, damage aura via Karthus'sDefile, and Lay Waste poke, and abilty to have a global presence via Requiem. However, he is very vulnerable to ganks if overextended.
If you're taking Karthus as a top laner on Dominion (between the Drill and Refinery), Wall of Pain can prove quite useful as an initiation tool for turret teamfights (especially at the Windmill), as it increases the damage all your allies deal while slowing the enemy. (If placed right, it can even reveal their brush!) By taking levels at 3, 4, 10, and the remaining two between levels 11-18, you can become frightfully efficient at helping your allies cap points. As such, taking Garrison to further improve your initiation can prove valuable.
When played as more tanky build, consider Rylai's Crystal Scepter as a top-priority item. Its passive provides slow for all your spells, but especially for Lay Waste, which you can use to kite enemy with terrifying effeciency.
On either lane for Dominion, Requiem has a great deal of utility. Not only can it be used to secure kills, but it can also slow via Rylai's Crystal Scepter, reveal all enemies via Hextech Sweeper, butter up enemies for your teammates from afar, or simply work as a global capture interrupter.
Try building tanky on Karthus as you will try to stay in the teamfight with your Defile on and spam Lay Waste.
Playing against Karthus can be extremely frustrating, as avoiding death with a small amount of health left is nearly impossible due his Requiem. If you find yourself having trouble with an enemy Karthus, buying a Banshee's Veil can offer at least some protection to his ultimate. If you are a caster and wish different/more protection, grabbing a Zhonya's Hourglass earlier than you normally would can save you on many occasions. If you are an AD-based champion and wish for some additional protection, a Hexdrinker or Maw of Malmortius can block 250 or 400 magic damage respectively from his ultimate if he takes you below 30% health. Beware however, those items aren't as reliable as Zhonya's Hourglass.
Elixir of Fortitude is a good counter to Karthus's ultimate during the early and mid game. It costs only a fraction of the other items which would normally be used to counter Requiem.
Always remember, especially during team fights, that Karthus can continue to cast spells while he's dead without the cost of mana. When engaging his team, kill Karthus first, then retreat out of his attack area before continuing the fight.
When in the midst of a team fight, it may be difficult to notice the attack animation of his Lay Waste. Be wary of this if you are playing a squishy character and/or have little magic resistance, as Karthus' damage output is steady and deadly.
Also, do not underestimate the tick on his Defile. It is DoT, so it may not be noticeable at first, but its damage can be as much as 160 per second, not counting resistances, on top of his damage from Lay Waste.
Do not underestimate Karthus' 1v1 potential. He may be squishy, but his killing power vastly increases when fighting stationary, single targets. If you engage him as ranged, try to stay out of his Defile and keep moving to avoid his Lay Waste. If you need to remain stationary, stay near other targets that will be damaged by his Lay Waste (e.g.: Allied minions or neutral monsters) to reduce his damage output. If you engage him as melee, you should either only do so if you have the level advantage or survivability to engage him in his AoE spells and live.
Some Karthus players will intentionally walk into a teamfight and die, This is rather a great strategy if Karthus is in a situation where he will die because he can cast his spells for 7 seconds after his death. Make sure that after you kill him, stay away from his Defile.
Laning against Karthus can be challenging if you are not aware of his Lay Waste. Without at least one movement speed increasing item (e.g. Boots of Speed) walking away from your minions can be costly even while watching for Lay Waste's if the Karthus is predicting your movements accurately. To avoid the doubled damage, stand near one or two minions. That way, if Karthus attempts to harass you with Lay Waste, he is either doing minimal damage to you and the minion or placing it so that you have more time to respond, as the AoE is only half as close to you. Standing in the middle of your minions is a less favorable way to reduce damage as the unit collision can more or less hold you in place while KarthusLay Wastes continually.