Why?

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Why the title? "Don't Underestimate Me!"
There's a pretty simple explanation to it, but I'm going to skip that for now & make it complex! People often say, "Oh I hate Karthus, he's just a kill stealer" "All he is good for is his R" and of course my favorite "What a lazy champ, without his Ulti he's nothing."
I got news for those who underestimate Karthus, he's much more than just an ulti. People are annoyed with his passive just as much. Putting those two small details aside, if you want to think Karthus is just a pretty face with a painfully annoying ultimate, you've got a lot to learn. I'm going to explain that all in this guide. Of course if you're too lazy to read my work, feel free to use the table of contents and skip around to the 'Lazy' sections I've put in.
I've added a number of spoiler boxes to make it so you can wheel through faster, and open up the boxes you actually want to read. If you don't like the use of spoilers, please feel free to comment and tell me why. I enjoy feed back, always.

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Lets do some quick math for the above build that is previewed.
Because of the fact that the preview builds don't show the entire final stats (reason being that they don't calculate in masteries, or things like Rabadon's Deathcap) I feel like I should show you what the end stats can look like.
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Start

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If you've just gotten the boots and 3 pots, your stats should look like this:
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At Level 10

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If you have Hextech Revolver, Tear of the Goddess, and Sorcerer's Shoes
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At 18

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If you have all 3 Core items (including Eleisa's Miracle)
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Full Build

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Using the Example of a Full Build above, most of these are the ones that change:
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WAIT

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How can that be? 444.85 to 711?
First off lets explain what isn't in this calculation:
24 pure AP from Blast & Mental Force
72 Mana from Expanded Mind
25% of your AP from Rabadon's Deathcap
5% Increased AP from Archmage
3% of your mana converted into AP from Seraph's Embrace

Lets start out with a simple conversion from Seraph's Embrace
3% of 2440 converts into: 73.2
Now we have something that looks like this
[(444.85 + 73.2 + 24)x1.05]x1.25 = 711.44
OR
(542.05 x 1.05)1.25 = 711.44
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The Point?

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Don't just look at the preview stats these guides show, make sure you know the math. Items and masteries aren't always calculated into that big preview that you're so use to seeing. A 250+ difference is quite big, wouldn't you agree?

Pros + Global Ultimate + Defile offers great mana sustain + Hard hitting combos + Very diverse character + Feared by many + So many Pros that don't even need to be listed, he's just that good Having a skill like Defile, that works in two ways, is always a great tool to have in your kit. The fact that it can grant mana per unit kill (when its off) really makes farming that much more easy for a champion like Karthus. Then he has the gift of chaining his abilities into a deadly onslaught to anyone in range (more detail in the Skills Section). He's so diverse in his unpredictability. Foes cannot know when he will pop up, or strike, he just does. Especially when you use tricks like Global Teleport. People know Karthus for his ultimate, but his power is in all of his abilities. If I were to go into further detail, & be more knowledgeable in these fields, I'd tell you all about jungling as Karthus. Unfortunately I have yet to try this, but I will someday, however that doesn't mean I think its bad. Quite the opposite, I've seen a Karthus jungle first-hand. So let me tell you this, its dangerous, brutal, & unpredictable. Just another reason to fear Karthus.

Cons - Very Squishy - Sometimes causes you to feel immortal - Often the target of ganks - Hard to master - Requires dependable teammates - Wall of Pain's cooldown feels too long(?) Throughout the entire game, Karthus is squishy, and that's just a fact. One you'll need to get over, so you can focus on the fact he is undead! Which also happens to be a cause of the 'Immortal Effect.' Although that's mostly due to using Revive, you feel like, even if you do die, you live a while afterwards, and then you can revive fast. Awesome, but wrong! You cannot get careless just because you can revive afterwards, Revive is there for a reason. That reason isn't so you can use it after you die, more info in the Summoner Spells section. When laning, you can probably expect many ganks, namely if there's a Lee Sin or Evelynn jungling, people don't want to see Karthus' ultimate get any stronger. I could ramble about dependable teammates but, simple enough, he's not a one man army, you can be as strong as you want, and still die in an instant without the proper allies. P.S. Does anyone else feel like Wall of Pain's cooldown is longer than 18 seconds?

Playing Karthus isn't so simple that you can be carefree about your rune choices. Remember, Karthus doesn't just become strong, he needs to build & farm.

• Greater Quintessence of Ability Power: The universal quint for APC characters, this gives you a strong (rounding up) 15 AP boost at the start, which really makes the game. If you read the 'Lets do the Math' section, Initial Start, you'll understand how big an impact 15 AP is on your start.
  
• Greater Mark of Magic Penetration: Essential for most APCs, being able to cut past any amount of magic resist can really help. If you want, you can think of it like this; with a good amount of penetration, you can render magic resist useless. However you view it, what's important is knowing that you need to have a balance, pure AP can easily be countered by build magic resist. On the other hand, if you were to focus building magic penetration, you won't find yourself dealing nearly enough damage.
• Greater Glyph of Magic Resist: There are 3 choices you can make here, going with these are typical for a mid lane Karthus because they give you decent amount of magic resist right off the bat. Which helps defend against most mid-lane opponents. Another path you can take for those who are confident in their skills, is Greater Glyph of Scaling Magic Resist. This works well when there are a heavy amount of AP dependent enemies. Finally the last path you could take is for the really risky/confident players. Bringing along Greater Glyph of Ability Power means more damage dealt, but more damage taken. You won't find yourself being able to take many hits from other AP users, so think carefully.
  
• Greater Seal of Armor: I pretty much use these for every character I pick. These work wonders in all situations, so I really don't want to spend time going into mass detail. You need armor, point enough to pull these out and start off with an extra 12.7 armor.

Teleport: There's a few simple reasons for putting this in your toolkit. Popping in to help secure a kill, or gank an enemy (utilize wards in areas that the enemy probably has yet to ward) for undetected attacks. In combination with Revive, you can instantly Revive, & jump to a location for whatever reasons you may have. Having the passive from the boots [Enchantment: Distortion]] can grant you a reduced cooldown as well. Global Teleport, is what I like to call the use of teleporting a heavily warded map. If you or your team wards constantly, it just makes that many more places for you to appear.

Revive: One of the least common spells you'll see at higher levels, Revive grants Karthus an extra special weapon. Alongside Death Defied, his passive, he extends his life for 7 seconds, but if this wasn't enough time to finish someone off, or Requiem just finished its cooldown, you can revive & act swiftly. Now the cooldown is a painful one, 540 seconds. Combined with Eleisa's Miracle & Mastermind you can cut the cooldown to roughly 389 seconds, almost 3 minutes less. Hopefully if you are bringing these two spells, you'll read the part below.

Usage of Spells

People seem to underestimate the power of Archangel's Staff/ Seraph's Embrace
This item is much more than just a nice barrier, or extra mana. In the final build above, of the 711.44 AP, 133.2 of it comes from this item.