What website is completely free, when it comes, to public records?

For what location and subject ?

There is no one web site that has all public records. You have to define what info you are looking for first, then maybe there will be a site that has the info.

If in the U.S. depends on what you are looking for. This site, just as an example, explains that some frequently requested items, and some recently released or recently declassified documents are available on their website without an official request. Some states have a similar site.
In some cases, you are given the OK to inspect records to search for specific documents that may exist, then request copies at a certain cost per sheet.
Some records are withheld for legally specified reasons and some are redacted. There is no "one size fits all" regulation that I know of.
One typical U.S. procedure:
http://www.state.gov/documents/organization/128321.pdf

.gov websites and the like would contain what you need , such as Public Access to Court Electronic Records

Here's a listing of such places FAQ about background checks | Online people searches & more