Hello and welcome to my first guild.
So guys, I rarely see well played karthus's, as many lol gamers think karthus is just about his 'e' spell, which is his aoe(sorry i dont remember the abilities names) and that's what i thought first but then I wanted to try leveling up his 'q' first, and the damge it can do to a gingle target is insane and the range is pretty good to a great harrasment at early game as it also has a short cd time.

For runes, use mag pentration marks,
flat ability power seals
and I do a mix of ability power glyphs and flat ap.
Flat ability power Quints.
The reason i go for flat is because if you can go crasy at early game then your late game should be great too.. and you'll get a nice amount of ap from the items build.

for mastries i go 21-0-9, grabbing all the AP in offence and mana regain utility.
[mastries]

Pros:
Karthus can easily control mid lane and fear his oponent and wont let them farm well if possible.
Using 'Q' as offen can make your team go crasy and go right away FTW.
with Karthus's ultimate, you're everywhere on the map if you can keep an eye on what's going on bot and top lane, u can easily grab assissts and kills and maybe save your team mates from death.

Cons:
Karthus is squishy as ****, so u shouldn't be inside the fights so much as everyone would focus the carry, but karthus has a great range.
another thing which is his ultimate veryhigh cooldown time, i think it's the longest cd in the game.

Grab ur 'q' first and level it up first.
lvl 2 grab ur 'e'
lvl 3 lvl up ur 'q'
lvl 4 get ur wall of pain
then keep levveling up ur q, rank up ur ulti when ever its up

Note: SORRY GUYS I STILL DONT KNOW HOW TO GET THE GRAPHS AND DIAGRAMS, SINCE IM STILL NEW, DONT JUDGE