While it is impossible to remove every trace of personal private information from the internet, there are imperfect solutions to the problem of online privacy and reputation protection.  You can remove large swaths of information from the primary website sources which are available on the web.  While these websites are not usually the same background check companies which are used by employers, and only certain information can be removed, it is still a very valuable service to an individual looking to maintain their online reputation.  These background check websites are really data mining companies, and they collect lots of information about you which you might not know they have or which might be incorrect (Alexander Wainberg was listed as his own brother for a while on one such website). 

Recently I became aware of two lists of how-to opt out of these websites (or limit the information available).  One list on reddit is good but there is another page stopdatamining.me which appears to have a better list.  As always, check through both lists but my primary recommendation would be the use of stopdatamining.me since it is likely to be updated.