Add your liquid dye and warm water to the bucket. Use a bucket that's taller and wider than your lampshade so you can completely submerge the shade, and add enough mixed dye solution to totally cover the shade when you dip it. Use the proportions listed on the dye's instructions, and mix thoroughly. For a stronger, brighter color, slightly decrease the amount of water you add; for a lighter, more transparent color, increase the amount of water.
Fill your spray bottle with water and turn its nozzle to the mist setting. Evenly spray water all over the lampshade until the shade's fabric is damp. Do not over-spray; if you accidentally saturate the shade with water, allow it to dry completely and try again.
Grab your lampshade by the top of its inner wire frame. Dip the shade into the bucket, and completely submerge it. Pull the shade out of the liquid, and check the color. Dip again for additional saturation.
Hang the shade outside to dry. Use a length of string to tie the top of the lamp's wire frame from a tree branch or clothesline.