To make a homemade cleaning solution for LPs, start with a gallon of distilled water and add one or two drops maximum of Dawn dishwashing detergent. Apply to the surface of the LP with a clean (preferably new) lint-free soft cloth. Rinse with distilled water to make sure all the detergent is off, avoiding the label so you won't damage it, then blot with a dry lint-free soft cloth or, preferably, if you have the time allow the LP to air dry by propping it up on its edge on a lint-free cloth. If you have a little more money, instead of the Dawn, add a photographic wetting agent (which you can buy from photo products stores or online) such as Triton X-100, Triton X-110, Triton X-115 or Monolan 2000, following the instructions on the particular product.
The Library of Congress recommends using a product known as Tergitol™ 15-S-7 Surfactant. Place 2 milliliters of Tergitol into a suitable container (glass, stainless steel type 304 or 316, fiberglass-reinforced polyester, polyethylene or polypropylene) and fill the remainder with deionized water. Then use any clean spray bottle and spray the solution onto the surface of the LP, wipe any excess with an eyeglass cloth or other similar soft wipe, rinse with more pure deionized water, then dry with another soft cloth. If you're using a mechanized cleaner, follow the manufacturer's instructions in pouring in the solution.
There are many premixed LP-cleaning solutions available on the market today. You might be able to find these in music stores, but more likely you'll have to special order them via the store or Internet. These include Discwasher IV, QED Record Clean (small spray can), LAST Record preservative, Genie-In-a-Bottle, RCA Discwasher, AIVS Record Cleaner Vinyl Cleaning Fluid 3 Step Kit, Phoenix Record Cleaning Fluid with enzymes and Gruv-Glide Record Treatment Kit. The links contained in Resources below have more information on these products.
There are a few manufacturers that have LP-cleaning machines or "vacuum cleaners," which are basically a box with motors which turn a platter containing a vacuum suction tube with an applicator pad or soft brush attached. You just place the record on the platter, prime the solution pump, then let the record spin for a few revolutions. Nitty Griitty, VPI and Keith Monks are three such examples of these machines. If money is no object, there is a laser turntable now available which includes a VPI cleaner but also uses a laser to "read" the record instead of a stylus, which can actually play many LPs that would otherwise have to be trashed.