I cut the sides from 1/4" balsa and the top, bottom, and rings from 3mm plywood, but the dimensions aren't very important.

Glue the sides to the bottom, the top to the sides, and the small ring over the top hole. Cut a piece of crinkly plastic - the cellophane wrapper around a box of teabags is perfect - and use the large ring to hold the plastic over the small ring. Sing or hum into the fat end of the kazoo, not the skinny end! If the plastic doesn't buzz with your voice, try making it a bit looser or tighter, or try a different kind of plastic.