The melee attacks are the ones you do without flames (swatting with your tail, etc.). You have to hit that specific button sequence (circle, circle, circle, X, circle, circle, circle) in the time allowed to move on to the next lesson.
Unfortunately there are no quick ways around this- I agree some of these can get a bit frustrating, I went through the same thing when doing up the review for this. All I can say is keep trying and you will eventually get through it.