I have made a few of these and the best design I have found is Larry and Lin Pardey's from their "Cost Conscious Cruiser" book. This has been published in article form in numerous magazines and online publications. My version is a little different - rather than sewing in a double bottom, I use a second boltrope around the bottom and then pound in a round wooden piece to serve as a bottom.
The first step is to make the boltropes. Use about 60 inches of 3/8 inch three strand to lay up 8 inch diameter rope grommets. If you use over 3/8, it may be hard to pass the boltrope under the needle of your sewing machine into the throat, but if you are hand-stitching 1/2 inch rope is fine.
You need two boltropes for a bucket. Start by unlaying a 60 inch length of three strand and taping the ends. Draw an 8 inch diameter circle on your workbench and place a single strand on it, with the two ends of equal length crossing at the top. Then start laying up one of the tails around the circle until you get to the bottom. Do the same going the other way with the other tail, so they meet at the bottom. Then cut half the fibers out of each strand and lay them around for a couple more crowns . You can whip the area where the two ends meet but since the whole loop is going to be covered in canvas there is no chance of it coming apart anyway. You end up with a two-strand grommet which is plenty strong enough but if you want you can continue around with both ends and make it three-strand. Just make your circle a hair under 8 inches or the extra thickness will make the grommet too big around to go into the bucket. If my pictures aren't clear, there are plenty of sites on the internet on how to make rope grommets. The important thing is to end up with one that is 8 inches in diameter. A single 60 inch length of three strand will make three grommets.
With a couple of boltropes made up, the next task is to measure and cut the canvas. You need a piece 27 inches wide and 21 inches tall. If you are using new canvas, let the serged part be along the top of the 27 inch length. If not, let your canvas be an extra 1/2 inch tall and fold it to the back and stitch. Once the bucket is done, all the unserged edges will end up buried.
I stitch boat canvas with V92 thread, which is very strong, with little stretch and good UV resistance. It takes a size 16 or 18 needle to handle this thread, and you don't want to use your home tailoring machine. Either an old depression-era Singer 201, or a Sailrite or similar machine is what you need. The new machines will slip their belts and burn up if you do much canvas work with them. Also, if you have done much tailoring or dressmaking, you may find the general rough and coarse nature of canvas work to be distressing. But it's worth getting used to, since a good canvas worker is worth having aboard a boat.
Fold the piece of canvas so you have a rectangle 13 1/2 inches wide and 21 inches tall. Run a line of stitches down the long folded edge 3/4 inch in from the edge, then a second line about a half inch in from the edge. Now you should have a long tube with the top and bottom ends open.
Next fold down the top edge all the way around to form a double-walled tube 10 1/2 inches tall, and then force one of the boltropes up between the layers to the top. Now it's starting to look like a bucket, right? Raise the needle on your sewing machine as high as it will go and remove the foot, and you should be able to slide the end of the tube enclosing the boltrope up into the throat of the machine. Stitch all around just below the boltrope, trying to get as tight up against it as you can. Run a second row down about a quarter inch.With the top rope sewn in, flip the bucket end for end and turn back the outside layer of canvas. Set the lower boltrope on the inside layer, fold the canvas over the boltrope and stitch it in place. Then pull the outer layer down, pull over the boltrope and stitch in place. You may want to run another row of stitches to lock it all down. At this point, the machine stitching is done.
A grommet needs to be added at the top of each side of the bucket, just below the rows of stitches holding the boltrope in place, so the bail can be attached. You can hand-sew them or use pressed in brass ones. The hand-sewn ones are more traditional but the pressed in ones are much easier and quicker. Then it is a simple matter of eye-splicing a rope through the grommets to make the bail.
All that remains at this point is to cut a round piece of wood to use as the bottom. I like using cypress because it is water-resistant. Cut it on the large side so that it is a tight fit. Spread some silicone sealant around the inside of the bucket, just above the lower boltrope, and then drive the wooden bottom down so that it seats all along the top of the lower boltrope. Don't expect to get a watertight fit, but it should be tight enough to draw water from overboard for sluicing decks.
One nice touch is to stencil your boat's name on the bucket. The best time to do this is early on, before you sew in the top boltrope, while the fabric is laid out flat. You should stencil your boat's name on the bucket so you have a better chance of getting it back if you are separated from it. It is also a convenience for the Coast Guard if they have to clear up the wreckage. Buy a stencil set or cut one of your own. I cut mine from .010 inch styrene. Some people say I have too much time on my hands. I say some people watch too much TV. Obtain a stencil set one way or another and make it a habit to stencil anything loose on your boat.
To the eye these buckets look small, but in fact they will hold two gallons of water, which is 16 pounds, plenty enough to dead lift over the side of a boat. Something fun to do is to make small buckets, 6 inches in diameter and around 6 inches deep, for children's toys. For most kids, it may be the only thing they have that is not made out of plastic.