This mission is fairly straight forward, Vegeta will become a great ape, and you must defeat him. Unlike a normal fight, you must stagger your enemy, as indicated by the bar beneath his health.
Normal attacks, supers and ultras can all help lower this bar. If you position yourself near his tail, you can also get a lucky tail grab that does massive damage to his stagger gauge. Once the gauge is emptied, Vegeta will be stunned and you will have a few seconds to do actual damage.
Once he reaches 50% HP, Krillin and Kid Gohan will join the fight. Repeat your previous process until he runs out of health to beat the level.