Most important thing is insurance. Try a firm called Cliverton, they do special policies for pet sitters. Other than that, its a case of advertising and getting people to trust you with their pets (and their home of course). Some knowledge of animal first aid would help, and a reference from your vet might be good.
I started a pet sitting business in 1989, well before it became a popular thing. I had previously run boarding kennels and spotted a niche in the market where people did not like leaving their pets. Only problem was people all want you at the same time and unless you take on staff you are stuffed for keeping them all happy especially at times like Christmas and Bank Holidays. Cats are easier (provided they don't escape if they are indoor cats) because you can visit several in one day and they don't need exercising like dogs do. Some people want you to live in and some are happy with daily visits.