Here's how to do it.
First you will need a plastic bucket with a lid and a handle. A 5 gallon bucket is recommended. One of the best is a 5 gallon plastic cat litter bucket. They come with a lid and handle and are made of thin plastic that is easy to cut.
Step 1: Cut four 2" diameter holes, spaced evenly, in the lid of the bucket (see image).
Step 2: Cut a 4 inch diameter hole on opposite sides of the bucket about mid-way down from the top rim of the bucket (2 holes total...see image below).
Step 3: Use 4 or 5 thicknesses of paper to pad the bottom rim of the holes cut in step #2 so the sharp edges do not cut into the tomato plant stems. Tape in place.
Step 4: Puncture three drainage holes in bottom of the bucket with tip of small knife. Just puncture..don't cut out any plastic.
Step 5: Fill the bucket to bottom of side holes with planting mix.
Step 6: Lay tomato plants on dirt inside bucket and guide the top of plants through the holes on each side of bucket (see image below).
Step 7: Fill the remainder of bucket with planting mix to within 2 inches of top rim of bucket.
Step 8: Place lid on bucket.
Using the bucket handle, hang bucket so bottom of bucket is at least 4 feet above the ground surface. Water from top of bucket allowing water to run through holes in lid of bucket.
The plants will grow up toward the sun at first. Then as the limbs grow larger and have the weight of tomatoes on them, the plants will hang down toward the ground.
Check the soil moisture every day for the first week. Do not allow the plants to dry out but, at the same time, do not allow the roots to sit in soggy soil for extended time. After the first week you will be able to judge how often to water by observing how well the bucket drains and how well the soil retains moisture.
Be sure to locate the bucket in your garden where the tomatoes will get full sun all day.