The intelligence committees in the Senate and the House (it's called a joint committee) oversees the FBI. The Director would have reported any potentially impeachable offenses to the heads of those committees. He would have also reported it to the Justice Dept. Since AG Jeffers removed himself from this investigation to avoid a conflict of interest, it goes to the Deputy AG. The Deputy AG is Rod Rosenstein, who happens to be a Democrat. So the fact that Comey did not tell them anything says that he didn't think Trump violated any law.

This is from the FBI's website:

The FBI’s activities are closely and regularly scrutinized by a variety of entities. Congress—through several oversight committees in the Senate and House—reviews the FBI’s budget appropriations, programs, and selected investigations. The results of FBI investigations are often reviewed by the judicial system during court proceedings. Within the U.S. Department of Justice, the FBI is responsible to the attorney general, and it reports its findings to U.S. Attorneys across the country. The FBI’s intelligence activities are overseen by the Director of National Intelligence.
https://www.fbi.gov/about/faqs/who-monitors-or-oversees-the-fbi

As for Trump's tweets, he just needs to stop. His big mouth (and fingers) just get him in more trouble. If he insists on tweeting he needs to have them vetted. The problem is that he's a known control freak and micromanages. (Obama was said to have been a micromanager too, but he studied law and knew when to shut up.)