The reward will be a gleaming lamp that will require only a little regular dusting and polishing when necessary.
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Fill your sink halfway with hot water and a squirt of dishwashing liquid. Soak one of the cloths in the soapy water and ring it out so that it is damp but not dripping.
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Unplug your brass lamp, and remove the shade and light bulb. Scrub and wipe all the outer brass surfaces of the lamp with the damp cloth. Replenish the cloth with soapy water regularly as you work. Don't worry if it doesn't look shiny yet -- you are simply removing the surface dirt and dust from the lamp.
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Dip the toothbrush in the soapy water. Clean any fine details or nooks and crannies on the lamp that you can't get to with the cloth.
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Dry the brass lamp with an old tea towel.
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Apply the brass cleaner, following any manufacturer's instructions. Usually, this means you wad up a dry cotton cloth, apply a small amount of the cleaner to the cloth, and rub the surface of the brass lamp. Start with a small amount of the product and add more to the cloth as you work.
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Rub the cleaner into the brass using small circles and back-and-forth motions. Apply firm pressure and work in small patches. The brass should become shinier and brighter. Continue until you have cleaned and polished the entire lamp.