What type of company
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You can find plenty of information on the web about setting your own business so the wheel would not be re-invented here. If you have no idea about the different types of business you can setup, there is alot of information on the website below
http://www.startinbusiness.co.uk/flowchart/4flowchart_legalformat.htm
Ok let's get back to the setting up your leafleting business.
It is possible to set your self up as sole trader, just pop down to the bank open a business account, it's that simple.
You can also have your company setup by companies that specialise in company formations. Prices of what they charge and the services they offer vary so you will have to do your own research to find which company has the best offer.
I suggest you start by searching for the term 'company formation' at Google http://www.google.co.uk/search?q=company+formations
If you are going to choose this route make sure you also get a bank account created as well.
Remember you don't have to setup a company as you can get paid in your own name and take payment in cash.