How could an American president abolish the FBI?
The president can't abolish the FBI without a law, which law requires a majority vote from both the House and Senate and the signature of the president.
Furthermore, the FBI has exclusive enforcement for some crimes. Each of those crimes would need to be reassigned to another agency or they become unenforceable.
The president can reduce the effectiveness of the FBI through appointments or lack thereof; but, until the law is changed, any future president could reverse the prior decision.
Presidential Reorganization Authority: History, Recent Initiatives, and Options for Congress, December 11, 2012
Between 1932 and 1981, Congress periodically delegated authority to the President that allowed him to develop plans for reorganization of portions of the federal government and to present those plans to Congress for consideration under special parliamentary procedures. Under these procedures, the President’s plan would go into effect unless one or both houses of Congress passed a resolution rejecting the plan, a process referred to as a “legislative veto.” This process favored the President’s plan because, absent congressional action, the default was for the plan to go into effect. In contrast to the regular legislative process, the burden of action under these versions of presidential reorganization authority rested with opponents rather than supporters of the plan. In 1984, the mechanism was amended to require Congress to act affirmatively in order for a plan to go into force. This arguably shifted the balance of power to Congress. The authority expired at the end of 1984 and therefore has not been available to the President since then. [Emboldening added.]