According to Stain Removal 101, dry cleaning solvents are quite difficult to find in local stores. There are however, companies that offer dry cleaning solvents for purchase online, such as Enviro Tech International, and Guardsman Furniture Professionals.
As Stain Removal 101 notes, dry cleaning uses liquid chemicals to remove fabric stains, contrary to what it's name implies. The U.S. Environmental Protection Agency describes the dry cleaning process as adding stained fabrics, a chemical solvent and a small amount of water to a machine, which then agitates the stained fabrics, much like a washing machine. Once the fabrics are cleaned, they are dried, shaped and pressed.
According to the EPA, Perchloroethylene, or Perc, is the most commonly used solvent in the dry cleaning industry, although other chemicals are sometimes used. The EPA notes that Perc is also used to clean certain types of metal machinery and it is a toxic chemical which has caused major environmental and health concerns. Because of this, the government agency believes it's important that any used filters which retain even small chemical traces of Perc be treated as hazardous waste. As noted in their fact sheet on Perchloroethylene in 2012, the EPA is working with the dry cleaning industry to reduce Perchloroethylene emissions and to possibly abandon the use of the chemical altogether in the future.