Hello! I am Zujewski on the NA Server, i'm currently in Platinum and have come here to share my knowledge about Karthus, who is by far my favorite mid laner. I started playing LoL in the 4th Season, and picked up Karthus as soon as I saw his rework, and have loved my little lich since. This is my first guide, so bear with me and my large wall of text.

Pros

Amazing Waveclear
Terrifying Ultimate
High Sustained Damage
High Burst Damage
Passive allows you to always contribute
Gets beefy with RoA/Seraph's Embrace

Cons

Early Game is Rough
Lacking Roam Potential
Needs to Farm Decently
Can be Bursted Easily
Has to Position Well for Fights

Karthus' runes are fairly simple - you can either go for straight MPen/AP Scaling/AP Quints/Flat Health or you can go for more defensive runes using Flat MR and Flat Armor, but I believe the AP Quints and the MPen marks to be mandatory to any real success.

Karthus' masteries are more focused on damage than anything else, however we do take the few points in Utility because they provide... well, utility! More useful than anything that would be found in the defensive tree anyways. Movespeed > One or two extra mitigated damage. It is better to not take any damage at all than to take a little less damage, am I right?

Flash - This spell is essential to practically any champion, as its versatility is just too great not to embrace lovingly into your cold, dead arms.
Teleport - I believe teleport to be the next best spell, as Karthus' strength comes from how well he farms and how quickly he can stack his tear, and teleport is just the spell to ensure you don't miss much farm, and that you can go to base and quickly be back in your lane with an edge on your opponent.
Exhaust - This spell is also viable on Karthus, but it's a matter of preference in my opinion, you can either reduce the damage one enemy team member deals or be able to constantly farm and put in tremendous pressure with your teleport being up. Both spells have their merits, but I prefer teleport.
Ignite - No, i'm sorry, but no. Karthus has little use for ignite unless you're smurfing and want to get easy early kills. His lane is not meant to get kills on the enemy laner, but to farm safely. Ignite is an aggressive spell, save it for champions like Zed or Leblanc.
Barrier - I understand how this spell could look decent, but I believe the usefulness to be outshined by both Teleport and Exhaust, this provides a small amount of versatility compared to the other spells out there.
Cleanse - I see the merits of this spell, but just as with barrier I believe it to be inferior to Teleport or Exhaust, if you get caught in the first place you're pretty much dead anyways.
Ghost - You will not be alive long enough after you use this spell to get into the enemy team for it to prove very useful, go with Teleport or Exhaust.
Heal - Eh,as with Barrier and Cleanse, it has its uses, but is not as useful as the other spells available.
Smite - This is not a jungle Karthus guide - though you could find those. So do not take Smite in a ranked match if you're going mid or top.

If there is one thing that Karthus needs to do well, it is farm. Without farming decently you will take a long time to reach your huge late-game power, thankfully we have one of the best farming tools in the game with our Q, which does double damage if it hits a single target, so you ought to aim it on the edge of the minion you intend to hit. You should learn the damage of each rank along with your AP to be able to best estimate if you'll be able to kill that minion with your Q.
In Mid-Game, farming is a simple matter of walking into the minion wave, pressing E, and doing "/d" while in said wave.

[*] Rod of Ages - I believe this item to be practically mandatory on Karthus, the stats it gives are just too good to pass up, you need the health, mana helps with your AP boost on Seraph's, and having more AP never hurt anyone - except on the other team.
[*] Archangel's Staff/Seraph's Embrace - This item should be purchased as soon as you can after Rod of Ages, as the power spike from Seraph's is massive, and you should have almost a fully stacked tear (600-700) by the time you can purchase this item.
[*] Sorcerer's Shoes - I honestly don't see much viability for anything other than Sorcerer's Shoes, with the all-important MPen you need to send the little carries on their team scurrying away with their meager MR, I believe this item to be better than any other boots.
[*] Void Staff - Now here it comes down to how the game is played out, if the carries and/or tanks have bought a decent amount of Magic Resist, this item is to be bought before Rabadon's, as the penetration spike allows you to laugh as the carries wonder exactly why 1/2 of their life is gone right as a teamfight begins.
[*] Rabadon's Deathcap - This item makes Karthus' late-game, as the power-spike it provides is just astonishing, not only does it provide 120 AP, but it provides 30% additional AP on all the AP you already have, which will send your AP rocketing up towards 600-700, and around 800-900 after you finish your build.
[*] Zhonya's Hourglass - This item is amazing, but it should not go before the other items in this build unless you're fighting something like a Zed. This item allows you to have tenseconds of invulnerability on top of the time you managed to stay alive, which is massive when your Defile is constantly active, dealing major damage to everyone in the area, leaving your foes to tremble when they hear the song of the dead coming from above as they flee the scene of the massacre.

[*] Lay Waste - Q : This is your primary farming/harassment tool in lane, and later in the game will chunk out 500-600 HP on a single target, it has a great range and is your best tool in lane to either farm passively or go harass your enemy laner. You max this first.
[*] Wall of Pain - W : This skill is your only "escape" as it has a decent slow attached to it, and later in the game you will want to aim this skill on as many people as possible, because it will leave them wondering why the Banshee's and Spirit Visage they just bought don't seem to help any - this has a built-in MR shred. You max this last, but grab a point at level four.
[*] Defile - E : This! This is what makes Karthus' damage so devastating in fights, not only does he have his 600-900 damage ultimate, but he's constantly dealing 150-250 damage per second to everyone in his radius, plus it regenerates mana for you every time you kill a minion with it deactivated, which is perfect for us in lane as we practically never have to worry about going OOM if we're farming properly. You max this second, grabbing a point at level two.
[*] Requiem - R : Now this is probably one of the most dreaded abilities in the game, a global-damage nuke that will tear your opponents apart in the later phase of the game and is great to pick up kills that your laners missed in the early stages. Despite all of this skills usefulness, it can be countered by a Banshee's Veil or a Zhonya's Hourglass, which renders the damage null and void. You max this whenever you can, which are levels six, eleven, and sixteen.

This is where Karthus' strengths really come into play, with his high sustained damage, high single target burst, a slow and MR shred, and a high-damage ultimate that can be cast after your death, there's nothing not to love. You're the kamikaze pilot of the Japanese, you go in after your tank, activate W-E and spam Q until you're dead, followed by pressing that R button and have either won the fight or done enough damage to convince them to back off.

Karthus' passive is by far the most unique one I can think of, with him not avoiding death, but denying it for another seven seconds after he has been destroyed, allowing free spell casting without mana restrictions or possible interruptions, and your E is automatically toggled on. This is the skill that really makes Karthus shine, as he is going to hurt if he walks into the enemy team and dies with his MR shred and his constant damage on his E, followed by his ultimate. Nobody will be alive to tell the tales of the now dearly-departed enemy team.

Overall, Karthus is a relatively reliable champion that after the first stages of the game dishes out massive damage that few carries - or even tanks - can handle, and even then he provides a decent amount of utility with a massive slow and MR shred, plus he has astonishing waveclear, and if you can land your Qs with some sort of accuracy, the enemy will be left wondering why they just took 700 damage from one spell.
Anyways, thank you for reading this huge bundle of text, and feel free to comment, vote, or leave suggestions. Goodbye, and I hope you've learned something about Karthus!