Runes:
If vs ad:
mpen reds, armor yellows, scaling ap blues, and flat ap quints
If vs STRONG ap early/mid (ex lb, diana, brand, cass, kat):
mpen reds, scaling health yellows, 4 flat 5 scaling mr blues, and flat ap quints
If vs WEAK.MEDIOCRE ap early/mid (ex anivia, ahri, lux, tf):
mpen reds, scaling health yellows, scaling ap blues, and flat ap quints
MASTERIES:
21/9/0 90%. you can look under my profile for exact mastery (one labeled "karthus") which i believe maximizes his dmg output and ones he uses best.
Other 10% go 21/0/9 (and get the mana regen mastery). This is important vs champions like morde, morgana, etc, which have very strong waveclear but very low dmg output to you. You just need to keep up with their push and not let them just push you into tower and then go roam or farm side lanes.
BUILD:
Start dorans ring vs 99% of matchups. I honestly cant think of a specific champion that I would start anything else. Except maybe top lane vs heavy ad with strong gap closer I would start cloth 4pots 1 mana.
You then build a tear, then rush either RoA or Rylais. This is honestly dependent on preference. I personally prefer rylais for the insane kiting it gives him and the earlier power spike, plus its build path is amazing. You then finish arcangel, build zhonyas, then finish with rabadons/void staff.
Early game you basically wanna farm up as much as possible. Contrary to what people believe, however, he also has some nice poke once you get some good practice with landing Q. The main thing to remember about Q is that it deals double the damage if it only lands on one unit. So if your enemy is outside their minion wave, try to Q them a few times (assuming you dont miss any cs). If they are inside the minion wave just poke them with is and push the lane.
Which brings the next point. Never try to let the enemy push you in. This is because of two reasons. 1. Karthus is terrible at roaming; whats even worse, if you're pushed in, thats a prime time for your opponent to roam . 2. Karthus has a horrible time csing under tower, and you have to now almost the exact range of q to be able to kill minions after tower shots (ex the if tower attacks a side ranged minion, if your Q hits more than one creep when you're trying to kill it will not finish it off).
Karthus doesn't necessarily have any strong counters. This is mainly due to the fact he is so safe to play. By far the "hardest" counter would be katarina. Long range poke, pushes lane relatively fast, you have no cc for her ult, strong roamer, hard to kill/poke out in lane because of her build path. I literally hate everything about this champion. You can beat an inexperienced kat with some skill, but a good kat will just roam when she sees she can't kill you.
Easy matchups for karthus are low damage early game, low-medium range, bad roamer champions. Heim, Lux, Anivia, Syndra.
Role mid game is to try and get picks. Rylais Q makes chasing people very easy. Also remember huge power spike at lvl 11 due to increased dmg on ult. Try to poke people before initiating. Also karthus takes dragons very easily due to isolated Q.
Late game once you get your core items you become a monster. Basically when team fights start just try to run in, W the backline, toggle E, and Q as much as possible. Zhonyas when you're in the middle. Then back off if team fight is waning. I've also been experimenting with a somewhat different playstyle where you build Luden's Echo and staying in back procing a lot of the item's passive due to low cd of Q before going in.
Unrelated side note, both your Q and W can be used to check brushes for enemies, etc. Q gives a small vision for a half a second or so. W gives vision of an entire brush if at least one of the big pole sides are in the brush.
