It can be done. The DOJ does this on a case-by-case basis. And FBI has jurisdiction to investigate some of these things. Bernie Sanders has suggested requiring all police killings to be investigated.
Most Constitutional rights extend to people against the states. So if your civil rights (due process, etc) have been violated, it's against federal law.
It's problematic to expect relief for any but the worst abuses from FBI or indeed from any part of DoJ. They're all cops, and they're not going to be eager to police other cops. A more effective organization would be completely separate, and probably staffed by attorneys who had never served as prosecutors.
Most Constitutional rights extend to people against the states. So if your civil rights (due process, etc) have been violated, it's against federal law.