This achievement takes skill, luck and patience. One minute seems impossible but there are ways to help out. Go to the Options menu, Settings then Play Options and do the following.
Turn animation speed all the way up.
Turn auto play on.
Turn auto finish on.
These will all make the game go quickly, so you will save those precious seconds. Now keep playing single player CLASSIC klondike matches, the achievement cannot be unlocked in voyage matches. Also be sure to use the button shortcuts to make things move more quickly, draws more cards and will move a card to an available pile. After a while you will be able to finish matches very quickly, hopefully one of those occasions will be under one minute.