Some of the good things about Karthus are that he can burst people down with his lay waste and can keep the enemy from escaping from the fights with his wall of pain. During the team fights if karthus has zhyonia's hourglass he can put up wall of pain walk in activate his defile and use his zhyonia's hourglass so he wont take any damage but the enemy will. Another good thing about karthus is his ability to farm with lay waste and defile. With karthus you want to let the minions get as low as possible then you activate your lay waste near that minion or how ever many there are. If their is a big wave pushing your lane you should constantly turn off and on your defile. He can also really help the team out on killing the main objectives such as dragon and baron.
Karthus also has many things that hurt him such as his lack of movement speed, the mana costs on his abilities, and how squishy he is. Also the part that makes people not want to pick karthus is his incredibly horrible laning phase.

Since karthus has such a ****py laning phase you have to get through it and the only way to change a ****py laning phase to a good one is farm and use every gank you get wisely. Constantly ask your jungler for blue buff after the first one. In case you didn't know the buff spawns every 7 min so the timings for the first few are 7min, 14min, and 21min. When late game comes around make sure you and your team takes every dragon and baron opportunity that you get. Make sure your jungler or support wards dragon and baron and get pink wards or oracles elixir to clear out enemy wards from those pits. In fights you can choose to be a front liner or a champion in the back of the fights I personally choose to be in the front so you can engage with wall of pain then you can use your defile and activate zhyonia's hourglass and when you die you can activate your ultimate(requiem) to help out your team in fights by weakening the enemy and possibly kill them.