Pretty standard stuff with Doran's Ring and pots.
Farm up and try to get enough gold for some first back items. Get AP if you need more damage. Get Catalyst of Aeons for sustain if you're in a hard-ish lane.

If you need health then just start spamming Q for up to 15 health per second. Also works with Defile for even more health regen.

By the start of mid game you should have ROA. Get this ASAP to get it stacking sooner. ROA is quite cheaper compared to the other AP items.

While ROA is stacking you seem to lack a bit of power. Get Lost Chapter for some mana sustain or go for the AP components of Morellonomicon.

Getting even the components for Morellonomicon gets you plenty of AP to be relevant. Morellonomicon is fantastic against health regen champs and pretty good all around.

Morellonomicon
Pros: Components give a lot of AP.
Plenty of mana sustain. GET 20% OF YOUR MANA PER EACH KILL OR ASSIST. That is fantastic!
20% Cooldown reduction so you can spam your Q's even harder. Also, your ulti comes off cooldown that much quicker.

Morellonomicon + ROA give plenty of mana sustain and AP so you won't be lacking the rest of the game :)

Tear of the Goddess: I don't like this item. Sure, Tear with ROA gives the most AP and mana in the game but it's soooo late game it's ridiculous. ROA take 10 minutes to fully stack and Tear takes even longer that it's just not worth it. You want to be useful by mid game.
Tear and ROA is for ultra late game. Since most games are dictated by the laning phase and finished by the mid game, Tear and ROA just don't seem to give you any relevance until fully stacked. And by the time they are stacked then it's usually too late and the game has already been decided.

Rylai's: The ultimate utility item for Karthus.
Makes him a bit beefier with the health and gives him more AP. But more importantly is the on-hit slow you get. It procs with ALL OF YOUR ABILITIES. Hit someone with your E and they are slowed. Hit someone with your skittle and they are PERMA-SLOWED.

Kiting is much easier with Rylai's and catching up to someone is actually possible now. Get this item!

Void Staff: Not much to say here. If they start stacking MR then this item is essential.
It starts being cost-efficient when enemy champions stack around 75 MR and higher.

Rabadon's Deathcap: Not much to say here either. Get this for a huge AP boost. It's a scaling item so get it as 4th item or later.

If you need defensive items then go for what I have listed up at the top.

I've found succcess with these items.
Thornmail for those right clicking champions. It returns damage back as magic damage. You deal magic damage so this item is pretty decent for you. It really hurts if you have some Magic Pen to boost it some.

Banshee's Veil:
Get it if the enemy team is heavy on the AP side. It's also good for it's unique passive, the bubble. It helps in preventing you from getting picked by that Thresh hook or Blitz grab. Plenty of Health and Magic Resist.

Zhonya's and Guardian Angel:
Zhonya's is Godly if you need to dive the enemy team and deal massive AOE damage. Your team then cleans up. Gives an ok amount of AP. Nothing too special. Doesn't give enough armor for it to really mean anything.

Guardian Angel for pretty much the same thing but just once every 5 minutes.

Don't get picked off.
Don't wander alone.

You can take Baron and dragons easier now. Karthus does A LOT of damage to those.
Try not to die but if you do, do so strategically. Spam Wall, your Q's and finish with ulti. Your team should clean up any survivors.
BAM
Teamfight won!

Karthus has a godly lvl 1 All-in.
If your comfortable with Karth then go all-in lvl 1.
This works because you can spam your Q's and outdamage the enemy laner.
Most all of the other abilities have like 5 seconds or even more for cooldown.
Take advantage of this and do major damage and possibly even first blood!

Lvl 2 is a different story. Once the enemy laner gets to lvl 2 then it significantly harder to trade.

Now the lane is matchup dependent.

When trading in the early lane, be sure to hit your q's AND USE YOUR AUTO ATTACKS. Yes, Karthus does have an auto-attack animation. USE IT in the early levels. It does a surprising amount of damage and unlike your q, is guaranteed to hit.

FARM, FARM, FARM.
Always make sure your farm never falls behind. Try to have the most farm in the game.
Farm is very important. Farm is life.
Farm gives you levels and gold.
If you want to get stronger then get farm and level up!
Seriously, you get more stats and more power by leveling up! GET YOUR FARM!!!!!

Roam if you can. But basically you want your R to do the roaming for you. Pay attention to every other lane once you hit 6.

Basically, become omniscient once you hit 6.

Try to get mid tower down ASAP. This frees up the map for your team and hinders the enemy team.
It's hard as Karthus though, I know. But try to knock down that turret ASAP.

Die strategically, in the middle of everyone.
If your team is doing an important objective like dragon or baron, try to die in the pit.
You'll do damage to the objective and more than likely get a couple of kills too!

Use ulti to engage or use it to finish off survivors. Use good judgment.

Finally, Karthus is a decision-making champion. If you have a good head on your shoulders, and your not ****ing ******ed, then Karthus should be able to carry and have lots of presence throughout the game.

HAPPY SLAYING!