Turn a dining chair over on its side. Determine the exact center point with a ruler and mark a pencil dot in the center of the base of each chair leg.
Choose a drill bit that is the exact width of the stems of the casters, or the size recommended by the wheel manufacturer. Drill a hole into the chair leg at each of the marked dots. Hold the drill perfectly parallel to the chair leg.
Remove the stem from the wheel. The two parts should come apart easily by pulling them in opposite directions. Insert each stem into a drilled hole, then tap into place with a hammer.
Push the wheel back onto the base of the stem. It should click into place securely.
Place one of the plate caster wheels over the base of one of the chair legs so that the edges of the plate are flush with the edges of the chair leg base. Draw a pencil dot through each of the four screw holes in the plate. Repeat with the three remaining chair legs.
Choose a drill bit that is a size or two smaller than the screws you use. Drill four holes into each chair leg at the pencil marks, holding the drill perfectly parallel with the leg. These are starter holes to help you insert the screws.
Place the first caster wheel plate on the base of one of the chair legs, matching the four holes in the plate over the four starter holes. Place a screw through one of the holes in the plate and into the hole in the chair leg, and screw it in about halfway with a screwdriver. Repeat with three more screws in the remaining three holes in the plate.
Tighten four screws all the way into the chair leg.
Repeat Steps 3 and 4 with the remaining three legs of each dining chair.