I'm writing a book

[Fulcrum]

And in it the characters use private detectives and hackers to find the names, addresses and other personal information of other users on a gaming forum they use.

This story will take place in the uk and in rwanda.

What laws are being broken by them doing this?

Thanks in advance.

[Buenchico]

Simply searching google, or otherwise harvesting information from public websites (including gaming forums) is perfectly legal.

Otherwise the Computer Misuse Act 1990 would be the most likely piece of legislation which your characters would fall foul of:
http://www.legislation.gov.uk/ukpga/1990/18/contents

Chris