This is my first guide and I intend to keep it kind of short. I won't be going too much into detail about the mechanics of each spell and all, but I will give you tips and tricks on some abilities and items.

I just felt like doing this guide because I haven't seen any like it, and I really felt confident about it after 20+ win streak in normal. I only recently started playing Karthus in ranked games because before the championship games, I was afraid that I would be denied by other teammates for taking him since he is rather easily counterpicked. Now I know this isn't exactly the case.

By the time I started writing this, I'm sitting at 1708 rating in solo ranked and it will most likely go up in the following weeks.
http://aijaa.com/v.php?i=009828251978.jpg

+Lots of survivability without any actual hp items. Spellvamp works great for Karthus.
+Great jungling potential, you can solo baron at level 16 if geared up to hourglass.
+Having Will of the Ancients will help your team ALOT.
+Having both Exhaust and Wall of Pain will ensure for some really easy Lay waste hits.
+Also having Exhaust and Zhonya's Hourglass can ensure you with enough survivability to survive lots and lots of situations.
(For example Tryndamere will counter you quite hard without either of these)

-Not much AP in the early games, which will sometimes make your ulti ( Requiem) seem weak.
-About 800 AP with full gear, which isn't the best you could get.
-Lacking some mana in early to mid game.
-Low survivability if you don't master kiting with Wall of Pain.

What summoner spells I recommend:
I tend to take Flash and Exhaust for my summoner spells. That is because I want to have enough survivability through Flash and to increase it I take Exhaust. Too many times enemies will misjudge your damage and jump on you while you have Exhaust available, leading to their death. Also with Exhaust and having any spellvamp items does awesome things to your survivability.

Also having Teleport is not a bad choice, but you will have to sacrifice some much needed survivability.

What summoner spells to consider:
Ignite - While I like Ignite's destroying power, I have found that it doesn't actually increase you dps more than exhaust does. (Easy lay wastes with reduced mr)
Ghost - If you really want, you can take Ghost alongside with Flash to provide even more survivability.
Cleanse - I wouldn't take Cleanse, but if you really think you might need it, be my guest and do, it's not the worst possible choice out there. ( Cleanse can be a lifesaver with champions like Veigar on the enemy team)

Don't take the rest, just don't.

The item build I'm using doesn't have much AP early on, but your survivability and utility will make your presence noticeable everywhere you go. This build is only viable when the enemy team hasn't got:
- Veigar
- Kassadin
- LeBlanc
(- Akali)

So if the enemy team has any of these champions, I greatly suggest getting Banshee's Veil first, before Will of the Ancients. The reason I put Akali in brackets is while she is deadly as hell to Karthus and one of his nemesises, you won't necessarily be needing additional survivability if you just keep your head cool.

I normally start with Doran's Ring as my first item, as it gived me the early HP boost That this build lacks. After Doran's Ring I try rush Tear of the Goddess for some much needed mana. Also after I have the Tear of the Goddess I usually buy 2 Health Potions and 2 Mana Potions too.

For boots I usually take Mercury's Treads. Why? because I get additional survivability from them of course. Usually it goes around 3 seconds to kill you while you are stunned, but with mercs, you get out of CC earlier, resulting in happier endings. (btw, Mercs grant 5 more magic res than Sorcerer's Shoes grant penetration.)
IF they however have only 1 stun, taunt or snare, I will get the Sorcerer's Shoes, but those teams don't come in very often. (Some champions alone make me nervous enough to get Mercs though - Ashe & Veigar for example)

Hextech Revolver costs 1200 gold and from there 900 to Will of ancients. If I get more than 1600 gold after I have the revolver and before the Will, I get the Needlessly Large Rod, because you can't know when's the next time you actually can farm that much money without going back to base.

After you get Zhonya's Hourglass done you should be sitting at some very nice survivability and damage. Next stop is Archangel's Staff and Rabadon's Deathcap.

You don't usually get to Rabadon's but sometimes it even drags out longer than that. If no one else in my team has Abyssal Mask I usually grab it. If they do, I grab Void Staff.

Good teamwork is vital in any game, and Karthus can contribute to it by initiating a fight with Wall of Pain.
The exact reason I max out Wall of Pain first before Defile is because of the massive movement speed, armor and magic resistance reduction it gives. It also has great range and width to cut off escaping/attacking enemies.

Always try to look out for your enemies in sidelanes after you hit 6. You never know if someone could go down when you weren't watching. Always if someone dies, check the location they went down at, often in aftermath there are low hp survivors and you can finish them off. Communication also helps with this.

Also don't save your ulti for kills, but to help your teammates. If you see a teamfight going on but can't make it there yourself, it is often a good idea to pop your ulti just in case.

Some of these tips and tricks are pretty basic, but I see too many times people just not doing any of them, so I want to list some of my favorites.

After you get Hextech Revolver, you can easily do both mid and jungle if your team hasn't got any jungler - Just kill off the creeps in lane as quickly as you can and proceed to jungle -> kill 1 or 2 camps -> back to mid and proceed as necessary.
You can solo dragon with ease after you get the Revolver. Especially if you have the blue buff with you.

Akali is a very nasty enemy for Karthus, and if you feel like you can't escape her while you can't do any damage back, you can reduce her harass on you by bringing Vision Wards with you at all times. Of course you don't want to randomly sprout them all over the place but rather put them to where Akali has just stealthed.

After you get Tear of the Goddess, you really want to minimize the time it takes to charge up. So always when you visit store or are sure you won't be needing the mana, pop a few Lay Wastes. That way you will be having it charged up in no time.

Remember to use your Wall of Pain and Lay Waste to scout out enemies. The Wall of Pain is a great vision tool but it won't show you if an enemy is in the brush. Lay Waste will do this however.

When harassing an enemy in lane - Try to hit him where you don't want him to be. So you will either get the harass out or you will screw up his farming.

When a fed Tryndamere pops his ultimate and starts whacking his sword at you, you really want to have both Exhaust and Zhonya's Hourglass up. First pop Exhaust on him and after it wears off, pop the Hourglass -- This way you get more hp back due to your spellvamp items. This also counts for Akali.

If you see a dangerous projectile coming your way, use your Zhonya's Hourglass to save yourself some damage.