Purchase a sheet of 1/4-inch acrylic that is larger than your backboard. A see-through sheet will show the old backboard behind it, which could be useful if you want to keep the regulation square behind the hoop. But if your original backboard is too beat up, buy an opaque white or gray sheet.
Lower your hoop and use a screwdriver to remove the hoop and backboard assembly from the support pole. Remove the hoop from the backboard using the same screwdriver.
Sand down any cracks. Use a large-grit sandpaper to scuff up the backboard so that the acrylic adhesive has something onto which to grip. Wipe dirt off the board with a towel and clean it with window cleaner.
Lay your sheet of acrylic over your backboard. Trace the outline of the backboard using a marker.
Cut the sheet of acrylic along the marks using an oscillating multi-function power tool, scroll saw, saber saw, jigsaw or band saw. Use a fine-toothed blade.
Sand the edges of the acrylic sheet down using sandpaper. Make it smooth to the touch.
Apply acrylic adhesive to the backboard, place the sheet of acrylic on top of the backboard firmly and let the adhesive cure overnight. Use clamps to keep the backboard in place as the adhesive hardens.
Screw your hoop back into the backboard, reattach your backboard assembly to your support pole, raise the hoop and play ball.