The bare outline of You Kill Me sounds like one of those insensitive Polish jokes of long ago: Did you hear the one about the alcoholic Polish hit man who went to San Francisco to dry out? That’s exactly what happens in director John Dahl’s deceptively deep comic character study, but the movie has so many competing dramatic strands running through it, it takes some work to sort them all out.

The most important story, the one it does best, is the love story between transplanted Buffalo, New York, hit man Frank Falenczyk (Ben Kingsley) and a wounded but resilient San Franciscan named Laurel (Téa Leoni), who come to know each other by accident. The other strands—Frank’s career as a mobster, the recovering-alcoholic plot line, the gang warfare between Polish and Irish in Buffalo, and his brief, wry stint as a mortuary assistant in SF—are done pretty well, but in the end it’s Frank and Laurel we really care about, because of what we see in them.

Like many another movie hit man, Frank keeps things airtight. He lives alone by choice and doesn’t allow anyone to get too close—after all, the people he gets closest to professionally are his victims. From the amusing opening credit sequence, where he ventures out into his snowy Rust Belt front yard to shovel off the walkway, swigging from his vodka bottle all the way, Frank’s weakness is downplayed with dry, matter-of-fact humor.

Kingsley, lean and bald and laconic, does a variation on his gangster from Sexy Beast with just the right added touch of helplessness. He may hate his life, but you’ll never hear him complain. He just drinks. No wonder he squirms when, after passing out and botching a hit, he gets ordered by the boss of the Polish mob (Philip Baker Hall) to take a trip to San Francisco and clean himself up at Alcoholics Anonymous.

Of all the people Frank meets in SF—fellow AA member Luke Wilson, the oddly abrasive Bill Pullman as the mob “babysitter,” the embalmer from the funeral home where he works as a cover—Laurel is the most complicated. Leoni’s Laurel lives alone, too. She’s been hurt in the urban relationship wars, but she falls for Frank in a completely believable way, helped along by the bright, chiseled dialogue. Leoni’s performance is one of her best—nuanced, minutely detailed, and emotionally mature. When she opens up to the possibility of Frank, both of them blossom. The irony is that meeting her makes him a better killer. Before he encounters Laurel, Frank’s inner life is a discussion on solitude and death. After Laurel joins forces with him, we could even imagine them pulling hits together à la Prizzi’s Honor.

Dahl (Red Rock West, The Last Seduction, Unforgettable) succeeds in transforming two essentially tired characters, the type of losers we’d typically glimpse passing through a movie in one or two scenes, into fully fleshed-out human beings who fall in love.

YOU KILL ME ( * * * )

Directed by John Dahl • Starring Ben Kingsley, Téa Leoni, Luke Wilson, Dennis Farina, Philip Baker Hall, Bill Pullman, and Marcus Thomas • R, 92 min • At the Osio Cinemas.