Build the support box
Cut the top, bottom, sides and inner supports from a piece of 3/4"-thick exterior-grade or marine-grade plywood. Dry fit them together, then join permanently with a weatherproof adhesive, either polyurethane glue or Type II carpenters glue and #8 x 1 1/2" hot-dipped galvanized screws. Countersink the screws below the surface then cover the holes with exterior-grade wood filler before sanding and painting. The screw holes on the top will be covered by the checkerboard and top trim members.
Build the checkerboard
Start by cutting the baseboard from 3/8" exterior plywood. This thickness, combined with the thickness of tiles and adhesive, matches the thickness of the 3/4" top trim members that surround the tiles. Since tile thicknesses vary, adjust your dimensions to fit your tiles by dry-fitting these parts together before settling on the baseboard thickness. Ideally, you want the top surface of the trim members to be level with the top of the tiles. If you have trouble finding a combination of ply and tiles that matches the standard 3/4" wood surrounding them, simply plane thicker trim members down to suit. If you don't have a thickness planer, the parts are small and easy to do with a sharp hand plane.
Because tiles vary in size, set the tiles first before cutting the tile baseboard to its final dimension. There are two things that will ensure a successful tile installation: straight tile alignment and consistent tile spacing. To align the tiles, attach two 3/4" guide strips in an L-shape to the 3/8" baseboard. These straight edges keep the first rows of tiles aligned and square. To keep the tile spacing consistent, insert 1/8" strips of wood between tiles during installation.
To adhere the tiles to the baseboard, use a latex-modified thinset mortar. I used a product called Super Flex manufactured by TEC Specialty Products Inc. Spread the thinset mortar onto the baseboard using an 1/8" notched trowel. Cover as much area as you can comfortably tile in five minutes. The angle of the notched trowel from vertical determines the height of the mortar ridges. What you're after is enough mortar to cover the bottom of the tile when pushed into place. Too much and it will squish up into the space between the tiles--too little, and your tiles won't be held securely to the plywood.
Let the tiles set for 24 hours before cutting the baseboard to size. Use a tablesaw, with the blade lowered so it doesn't quite make it through the top of the plywood. This prevents the blade from being damaged by the tile or mortar. Use a sharp utility knife to complete the cut through the baseboard.
Next, centre the tile-covered baseboard on the project top and fasten it with glue. Make sure any glue squeeze-out is removed when doing this--hardened blobs of glue will interfere with the installation of the top trim members. Gluing large flat parts together like this can be tricky because they slide around. You can eliminate this problem by hammering three or four 1" finishing nails part way into the project top, then nip off the nails with wire cutters so only a tiny tip of metal remains above the surface. This trick prevents movement of the tile base board completely. Place a heavy weight on top of these parts before setting them aside to dry. After 15 minutes, check the edges again for squeeze-out and remove any while it's still soft.
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