This trophy simply requires you to play 30 games of VS Battle or Co-op.
Both modes are for local multiplayer so you should be able to get these just by plugging in a second controller.
For boosting this trophy VS Battle is the best choice. This mode does not have any goals and can be completed by just letting the timer run out.
As a side note, you should play through each of the Co-op stages if you are trying for the Completionist (requires you to collect all items) or the Enormous Katamari trophies (2 million Km katamari) as they create planets needed in your collection.