They would first eliminate possible causes, keeping in mind that most crashes have several causes. They would try to determine whether the plane broke up in flight and whether a fire occurred either in the air or after the crash. One of the first priorities for recovery is the propeller and engine. Hitting a body of water at high speed is almost like smashing into concrete, and a metal propeller such as the one on Kennedy's plane will bend in ways that can tell investigators whether it was turning or stopped. The engine is the most rugged part of the airplane, and it likely survived the crash largely intact.
Investigators can often spot an engine problem from an external inspection.
In high-profile cases such as this one, the engine is often sent back to the manufacturer for a detailed breakdown. Investigators can tell a lot by examining breaks and tears in the plane, sometimes by looking for paint smears or deep scratches on the surface that would be caused by parts that broke off the plane and hit other parts of the aircraft. Similarly, there are telltale marks for bending, torsion and shears. A fire in the air often leaves relatively clear marks. Even if ocean water washes away soot, a fire can leave unmistakable signs.