Cliffhangers can only occur on the following stages..
-Flow (Though it requires more work)
I'm going to assume you're doing this on Scramble.
If so, hit your opponent on the barricade twice.
Upon the second hit, they will begin to fall off.
If your opponent holds on, follow up with the "X or Triangle" button.
If that attack isn't blocked you will earn the trophy.
This can be boosted locally with a friend/second controller.
Set the A.I to the lowest possible difficulty setting to make things easier for yourself.