I saw two of his films recently:
The Heiress - Based on the novel Washington Square, the film takes place mostly in a townhouse in New York's Washington Square. Clift plays a guy who squanders his inheritence, and has no apparent goals or motivation in life. He proposes to Catherine Sloper, a woman who is set to inherit a large sum of money. Catherine's father doesn't approve, and believes that Clift is marrying her for her money.
I Confess - Set in Quebec, Clift plays a Catholic priest who hears a confession from a man who recently killed someone. Circumstantial evidence ties Clift's priest character to the murder, and he becomes the primary suspect and sent to trial. The priest doesn't wish to compromise the sanctity of the confessional, thus he keeps his mouth shut.
Both are above average films, and I recommend both. Available on Netflix. I Confess is especially good for the shots of old Quebec. Peter Bogdanovich provides some interesting commentary as well.