.. How do I get into the kitchen?

If you try to go into the kitchen while the cook is there, he'll kick you out.

If only he weren't there.

Open the door, and, after he gives you the metaphorical boot, walk to the western half of the SCUMM Bar.

Soon, the cook will come out of the kitchen.

While the cook is on the western side of the SCUMM Bar, scurry into the kitchen.

Don't worry. He won't walk in on you.