We are running online services and a disgruntled user of our's is sending false DMCA complaints to our providers. We would like to file a police report and take legal actions towards this individual
Welcome to the world of the internet. Unless you can show loss which is conclusively attributable to the disparaging remarks of your disgruntled user, you really have no recourse except to respond to the negative comments.
If you know the identity of the person, try to resolve the underlying problem to make this stop, or send a cease and desist letter. A lawyer can also investigate a tortious interference claim.