To do this you will need a plastic container that fits under your sink, or any place convenient for you. Put a number of holes in the bottom for drainage. It will need a tight fitting lid and a tray underneath to catch drippings.
You also need some red-worms. They do the job of digesting your kitchen plant waste, egg shells, coffee grounds and tea bags (Buy them on the internet or get a few handfuls from a composting friend).
Many commercial versions of the worm bins exist. Ideally they have more than one level as your worms will travel upwards as they finish with the level below.
Worms will keep working their way up to the new additions of plant matter while leaving their 'poop' called worm castings behind them. This is the richest soil there is-often called Black Gold.
You can take the 'black gold' out and mix a couple handfuls into the soil while you plant or just top-dress your plants buy spreading 1/2 - 1 inch evenly around.