Two things I'd like to add to this.
One, just don't overweight the needle, as this can wear out the vinyl prematurely dulling the sound.
Two, toilet paper can leave remnants in the grooves, which is what you're trying to eliminate in the first place. A soft felt-type cloth is preferable.
As for cleaning solutions, commercial record cleaning solutions are mostly alcohol. Use Isopropyl 90% alcohol or higher. Do NOT use rubbing alcohol or witch hazel as they contain additives that can harm your records. Dishwashing liquid without any moisturizers work well such as Joy or Dawn. It cleans dirty records well, but you have to avoid getting it on the label and rinsing all of it off quickly. Try not to get the label wet. If it does get wet then blot it dry immediately. Do NOT rub it. Try not to use tap water to clean your records. Instead use distilled water, which can be easily found in your supermarket or drugstore. The real trick is to properly rinse all this stuff off your vinyl records and get them dry without scratching them.