☛ LIFE hosted by Google: Ernest Hemingway by John Bryson, Ketchum, Idaho, February 1st, 1959. First published in LIFE magazine, vol. 51, no. 2, July 14, 1961, p. 68. © 2012 Bryson Photo.

The original captions from LIFE magazine reads:

At 60 and still full of the Old Nick, Papa Hemingway booted a beer can high in the air along a Idaho road. This was, he said, “the best picture I ever had taken.”

Ernest Hemingway committed suicide in the early morning of July 2, 1961. The LIFE magazine issue in which the above photo appears was dedicated to him (a portrait of him illustrates the cover as well). Hemingway and LIFE enjoyed a long relationship. It started in 1937 when the young Hemingway wrote captions for a photo story about the Spanish War and lasted up until the summer of 1960 when LIFE published part of his last assignment for the magazine The Dangerous Summer (about bullfighting in Spain: an augmented version was published as a book in 1985 under the same title). In the editorial for the July 14, 1961 issue, one can read: “(…) he was both a contributor to LIFE and a good friend.” (p. 2).
The photo was taken by photojournalist and long-time LIFE collaborator John Bryson. I wrote about Bryson a couple a months ago: he’s the same photographer who took the photo of Ingmar Bergman examining the shark from the movie Jaws.
Photograph Tim Keller shares some anecdotes related to this famous on his blog (see “Kick the Can” November 28, 2010: follow the link and scroll down to get to this entry). Apparently, it was Bryson who asked Hemingway to kick the can. Bryson also offered a signed copy of the same photo to his friend the filmmaker Sam Peckinpah.
There are some discrepancies online about the date the photo was taken. Time&LIFE Pictures (managed by Getty Images) has it taken on January 1st, 1961 at 12:00AM (the photo is filed as image #71934665). This is most likely eroneous for at least two reasons. First, obviously the photo couldn’t have been taken at 12:00AM. Second, on early Januray of 1961, Hemingway was just released from the Mayo Clinic (in Minnesota) where he had received quite rough treatments. His biographer Jeffrey Meyers wrote in Hemingway: a biography (New York: Macmillan, 1985):

[Hemingway] had hoped to return from the Mayo by Christmas 1960 but was released, in ruins, on January 22, 1961, and flown back to Ketchum in a private plane. (p. 550)

Corbis on the other hand has the photo taken on February 1st, 1959 (see photo no. 0000360469-006). I wrote to John Bryson’s son, Scott Bryson, to ask about the photo and he generously answered back, confirming that the date provided by Corbis was more likely to be accurate.
For more about John Bryson, see his Facebook Page (created and curated by his son Scott) as well as his official website and blog.