the worst matchups for karthus is jungles with high early game damage that are liable to invade you, if your in this situation is when i would advise getting the Trackers Knife and litering your jungle with wards also by smiting wolves.
on the other hand any passive jungler that just farms early you should be fine with.

Benefits of Karthus Jungle

Has amazing late game potential
Cam sustain early on without much problems
Great early game ganks with his wall

Disadvantages of Karthus Jungle

Can be invaded really easily
doesn't provide as much early game potential
Low mobility

You should always start with gromp and blue first for the most effective clear. without blue buff karthus will have a lot of mana problems especially if you use his e that drains insane amounts of mana early on and this also allows for karthus to have a isolated target with gromp so that his q can crit and do more damage. Early on with karthus its mainly about farming it up til you have your ult, after you get it you can coordinate with you team for some sick ganks!

With Karthus it is essential that you have your eyes on the minimap watching your team at all times and to make sure that you dont waste your ult. what i like doing is while walking anywhere if i see my team closing in on the enemies or the other way around ill have my ult ready to be of assistance.
Make sure you use the most out of his passive by getting off as many q's as possible and to put down your w before ulting, but also make sure you dont run out of time!

Playing Karthus jungle isn't a very serious pick but to me it is incredibly fun to play and makes for less boring games. i probably wouldn't recommend this in ranked play and say this is more of a pick for normal's and playing with friends. I hope you found this guide to be useful and good luck on the rift!