PROS
Karthus is a damage machine. If you get ahead you can expect to do double your ADC's damage. You don't have to walk out of lane to gank someone and you can get through laning phase without fighting at all.
CONS
Karthus has no mobility spells and rely very much on map awareness to steer clear of ganks. If you get behind too much you won't be able to do much. Your mind is constantly on alert when playing vs Katarina or Irelia or Zed. Laning will be stressful.

Karthus relies on getting his items fast to do well in the game. Because you are very immobile early game you will get ganked and poked alot if you don't position well. TP let's you get back into lane faster.
If you're vs an assassin and you're not very confident in your ability to cs from far away and position well exhaust can help you survive laning phase.

I prefer dark seal over doran's ring because you'll be able to get stacks pretty consistently throughout the game if you don't die alot. Having more mana is better than having more mana regen.
Lost chapter will carry you through laning phase. Luden's is almost always my first item. It's extremely good for waveclearing and synergizes with your ult. If you're ahead the first teamfight after you get rabadon's will be an automatic win.

Invading is fine. Karthus is very strong level one because he can use his Q a lot more than other champs can.
When the first wave comes make sure you get lv 2 before or at the same time as the enemy laner, but don't push the lane too hard or you'll be very gankable. You can trade lv 1 vs melee champs or champs with low damage early.

Don't stay in lane if you're under 150 hp. Try to have at least 1000g before you back. If your jungle is fighting over scuttler you should try to help. BUT if the wave is pushed to your tower DO NOT go help your jungler. Always keep at least 1 side of the river warded. Pinging mia is also very important because your roaming potential is a lot lower than most mids.
DO NOT USE YOUR ULTI BEFORE YOUR FIRST BACK UNLESS THEY'RE 100 HP OR LESS
If the enemy laner goes B and they don't have TP and their jungler is elsewhere on the map. Push the wave in as fast as you can and try to get a turret plate.

Your Q does double damage if it hits only one target. Use Q on a single minion by having it on the edge of your Q (you'll get most of the mana back). This is how you'll cs most of the time. After having 3 pts in your Q 3 Q's will kill the casters. After getting Ludens you only need 2 Q's. If you almost lose a cs just toggle on E for a brief moment and then turn it back off.
Don't use your autoattack to CS unless you're very good at timing them.

Your first priority is waveclearing mid and make sure they don't get the mid outer tower. Help your jungler take dragon if it's low risk because your Q kills it pretty fast. Take raptors whenever they're up because they give decent gold and you kill them really fast. You want to have a control ward in river somewhere. Always push out the wave before you roam elsewhere.
Splitpushing with Karthus is high risk high reward. You can aid the rest of your team if they fight at baron while you push bot. You'll get a tower out of this too. But the downside is that a splitpusher is traditionally a strong 1v1 champ like Jax or Fiora. If you don't have good map awareness and vision control you could just die for free making it a 5v4.

Your W has good pick potential because of its high range and the slow and mr shred. Put it right in front of anyone that's far from the rest of their team and you'll be able to get a kill or a flash. Toggle your E on immediately when engaging or being engaged upon because you'll either get a kill and get the mana back from presence of mind or die and not have to worry about mana.
CAST YOUR ULT AFTER YOU DIE OR IT WILL GET INTERRUPTED

IF YOU THINK THE GAME IS ABOUT TO END (win or lose)BUY ELIXIR BEFORE FINISHING YOUR ITEM
Karthus is a lategame monster because a single Q could chunk carries and your post-death damage is through the roof. They will engage on you every time so as soon as they do toggle on E and turn on Zhonya. The 2.5 seconds of E damage makes a big difference in lategame fights. Once you die Q a couple of times to try and hit someone then ult.