I do this at least once a week; try to do it twice, if I can, at the club I belong to. Try a damp micro-fiber cloth with a touch of liquid soap blended into it. I use a hand soap that has no additives like a conditioner in it. The pool tables I clean show minimal signs of burn marks or rail tracks.
I also vacuum with a brush attachment for the table area and a thin nozzle under the rails before I use the damp cloth. We have a small vacuum dedicated to this job. The cloth I use is lite colored cloth about 12x12 inches which I fold into quarters. I rub the cloth firmly; amazing how much chalk it picks up and also how much of the burn mark will disappear. I go over the burn marks/break areas and the rack area a little more aggressively because those areas show more use. Do the rail areas inside the pockets too; they get discolored due to the pounding they take. I'm gentler with rubbing down and vacuuming the rails just enough to get the chalk marks off.
I hand wash the cloth in our bathroom sink with the water running slightly and a bit of the hand soap until the water runs clear and ring it out good and hang it someplace to dry. The blue dye will stain the cloth; you won't get that out. I do two tables in about thirty minutes or less. The tables are dry in less than fifteen minutes.
Don't scrub over the spots; the dye in the spot might bleed out into the cloth and won't come out. Don't vacuum over them either.
To help reduce burn marks, especially while practicing jumps or breaks, use a piece of cloth under the cue ball. I've also used a couple of layers of paper towels to help minimize the marks.
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