Gloria Grahame in Naked Alibi 1954

Questioned as a murder suspect, solid (but drunk) citizen Al Willis attacks his police questioners, is beaten, and swears vengeance against them. Next night, Lieut. Parks is murdered; Willis is the only suspect in the eyes of tough Chief Conroy, who pursues him doggedly despite lack of evidence. The obsessed Conroy is dismissed from the force, but continues to harass Willis, who flees to a sleazy town on the Mexican border. Of course, Conroy follows. But which is crazy, Conroy or Willis?