Eric Lichtenfeld in Slate performs a close reading of the Die Hard catchphrase:

When terrorist-slash-exceptional thief Hans Gruber (Alan Rickman) taunts hero John McClane (Bruce Willis), "Who are you? Just another American who saw too many movies as a child?" and asks this "Mr. Cowboy" if he really thinks he stands a chance, McClane's answer—"Yippee-ki-yay, motherfucker"—marks the moment that McClane, an everyman, assumes the mantle of America's archetypal heroes: Roy Rogers, John Wayne, Gunsmoke's Marshall Dillon, and others who have been so vital to American boyhood. Unlike the many action-movie one-liners that are rooted in the hero's narcissism, McClane's stems from our collective wish-fulfillment. He is not referring to himself, not suggesting an "I" or a "me" but an us.
This is all valid, but I don't think Lichtenfeld gives enough time to the line's rhythmic beauty. Phyrric-spondee-trochee-trochee: a soft opening ("yippee"), a triumphant double-stressed cresting ("ki-yay"), and then the happy landing on the metrical regularity of alternating stressed/unstressed syllables ("motherfucker"). It's the rhythm that gives the phrase its playful feel and makes it more fun to say than "I'll be back" or "I'm your worst nightmare." Although the combination of gibberish and profanity doesn't hurt either.