Not to brag, but I was notoriously good with Kirby. My friends hated me :P Its been a while since I've played, so forgive me if I don't know exact key mappings.
Kirby's strength is not from his power moves (the B+direction ones) but from simple aerial attacks and smash attacks (the A+direction ones). The "drill" attack is your main weapon. Float around the map until you find a target you can drill into the ground. If your opponent sees you circling them, they'll counter the drill attack with a power move. Anticipate whether they're prepared or not, and if they are prepared try your B+up move.
After you drill someone into the ground, they'll bounce up for a few seconds. This is a great opportunity to use your A+up. This will knock weak enemies completely off the map. If not, they'll still be vulnerable in the air. You could just jump up and A+right them for a painful 2-smash combo, which will surely send them flying far away from you. Or you could try B+upping them or drilling them again.