You essentially need a solid base for you tile that does not expand and contract/flex or else the grout lines will crack over time. You can have a square or circular tabletop base cut to size made from 3/4 inch thick exterior plywood. You can use metal angle brackets to attach the top to the barrel. Cover that with an appropriate 1/4 inch thick backerboard, they can be the ones made from cement and mesh, cement fiber, or the new easyboard foam-core ones. Some say you can just tile on the plywood but do the job right. Apply a layer of thinset to the plywood, lay on the backerboard and screw in with the special backerboard screws as per manufacturer's directions. When set, you can go and lay your tile or mosaic.
You can band the edge with wood strips to finish it off or else tile it. You can also use the various cap shapes if you want to spend extra on the special tile.
Plenty of stuff on the web on how to tile and grout. Easiest to rent a wet-saw for a day or two to cut the tiles to fit the circular shape or else make do with a manual tile cracker for only straight cuts. Good luck.
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