First of all You should know that a strong caster and if you want to play him You should know how to aim :P. but really good at mid and the best ganker with his he can really hot hard if you play well with him.

Pros

-High damage in early mid and late game.
-Awasome poke but not as good as 's
-The makes U to kill after die awasome in team fight and easy revenge

Cons

-Low health early
-Hard to aim with (u can learn it quickly)
-if you get a stun or something hard cc u can pray to survive

Runes

Marks: I take magic peneration for marks its very powerful until the end of mid game after it the tanks almost have 200 magic resist and if you want kill them easier then buy its one of the best runes plus item combo aganist tanks.

Seal: I take mana regen its one of the best runes if you want first blood cause u dont need to take Defile to spam and with the the first blood is given.

Glyph: ap always good and not /lvl cause early game ownage with and endgame ownage with and so better if U make Ur more powerful in early

Quintessence: Ap again ofc

Masteries for more ap and more damage dealing and mana regen i think this is the best for
The 21-0-9 for high magic peneration and high ability power in early game and late game ater you buy the mana regen good for every caster which need mana :P so just pick it.

First the shoe ofc shoe :D after it to stack mana u can really fast stack up 1000 mana but start it early it is good, After build up the shoe too hit hard with after get for cc and if u get someone with U will get 300 gold for kill. After u should have the best and the most common item for ap caster after it u build up I think and much better then u got cc incredible mana little hp so just do it :D After i have the main ap i wanted i wanna deal some damage so buy best item aganist every champ Ur will hurt SOOO hard and ofc in late game u want to use ur ulti in every second but U cant do it so just but a That's all.

I take first for the first this is the main damage source from karthus easy to kill easy to farm with this. After i take for first blood (if fb already gone then fb on the lane). After just max after ofc and this give U hard slow in early in end game awasome damage.

I know lot of people who dont know when the worth to use because the big cooldown I always use when I that my friends want to takedown the enemy on other lain and everytime you see that u can have double and thats 100% then you can use it I almost had penta after a teamfight but have saved her and a teammate's life with

I take Flash and ignite.

one of the most common summoner spell good for taking advantage or run.

ofc for killing kathus need it if an ad carry has big lifesteal and you can kill them easily with the debuff.

if you dont know how to use and you run out mana and your jungler need mana and he/she doesnt give you the blue then it's good .

Farming is the easiest thing with karthus in early you just use to stack thats called double farm get gold from minions and get mana.

In late game this is changes. Easy farm now the you just go to the lain into 500 minion turn it up down and u've just killed 500 minion.

the first one is his such an annoying skill. I mean u can make him to 100 hp he just use it on minions and he come back with full hp and good to evade the

the second one is he can just jump out of battle with his and I don't know that you know it but he can evade with his so he is annoying too.

the third one is She has silence she has nuke in early she has she can be hard enemy in early so if you see her just call a gank as soon as possible the best ganker aganist is
He just come in with and taunt with Puncturing Taunt
so cant do anything.

I don't usually play jungle with karthus :D but in mid and late game you need the blue and you can do it alone just kill the little ones with and kill the golem with