There's no statutory requirement for a CRB check to operate a dog-walking business. (Indeed, since you can't get a CRB check on yourself, getting a CRB check would be impractical anyway).
Public liability insurance would be strongly advisable but is not a legal requirement.
No licence is required in general, but many local authorities now require people who walk more than 4 dogs at a time to hold a licence if they take the dogs onto designated public spaces. For example, see this byelaw which applies in Wandsworth:
http://www.wandsworth.gov.uk/Home/Environmenta ndTransport/Dogs/dogwalking.htm
Chris