You are pitted against a Giant Yoshi. All you have to do is simply kill Yoshi. Since Yoshi is big, it will take a lot of damage to kill him. Giant characters have a disadvantage though. He is a big target, easy to hit, has reduced jumping distance, and falls faster. Use smash attacks and items to knock Yoshi off the stage. Often times when you fight near the edge, Yoshi will kill himself while trying to attack you.
Easiest strategy
Use Jigglypuff. When Yoshi comes toward you, use rest. Since he's so huge, you most likely will hit him, and he'll just drop off the edge. you can get records less than 00:00:52 this way.