If you are out for kills early game I wouldn't choose Karthus. Just grab a nuke and have at it. But Karthus excels at farming. I would argue that Karthus doesn't have a weak early game in that his CS will be high, his low cost Q w/ the mana return of E makes him an excellent farmer. The range means you're out of harms way and at worst you'll be dodging skill shots so be sure to take boots to start with. If I see someone take ignite / flash, I know they will be wasting a lot of mana trying to push me out and make use of that aggressive combo.

Once you hit 6, make sure you have blue and watch other lanes for your R. Once you have blue, you are unstoppable farming machine. Push mid back and gank other lanes. Farm every piece of jungle on your trips and just have at it. A bad game for Karthus should be equal CS with mid and not having the oppurtunity to use R.

wen the ap carry on their team cant be super aggresive and deny your farm.
also if your team picked 4 ad champs because karthus will force them to not stack armour because aoe is da best.

Karthus Q range makes zoning him almost impossible and kinda hard? Karthus is countered by high burst, I tend to run flash and exhaust for summoner spells. Play him vs. squishes or /w a heavy AOE team.
Lvl one Karthus is strong too imo. The amount of support Karthus can give with his ult to side lanes is not to be underestimated as well.

He's harder to use against Janna, Lee Sin, Shen, Udyr, and anybody I'm not thinking of with a quick high-hp shield. If they can deny your ult, and have a good jungler, you become food.

Whenever the other team doesn't have soraka, and a really powerful burst/lots of poke mid like leblanc or ahri or even kennen

When they don't have these Karthus is a good pick. Oh janna too, her shields annoying....

I see, I was noticing that i never could win my lane, but I realize now that karthus isn't the type of champion that can do that.

So basically, when I play karthus, just farm like crazy, R when people need it, and melt face in team fights because of the amazing passive.