This is not really a card game; it is included here so that if someone proposes to play it with you, you will know what is involved.
Two players and a pack of 52 cards are needed. The first player offers the second player a game of 52 pick-up (or 52 card pickup). If the second player accepts, the first player scatters the 52 cards over the floor and says "52 - pick up!". The second player's job is then to reassemble the pack of cards by picking them all up.
That is all there is to it.
Wei-Hwa Huang suggested that the name may be related to Elmore Leonard's 1974 mystery novel "52 Pick-Up", which was made into a movie in 1986. However, the game existed long before this. Bob Durbeck reports having fallen for the 52 Card Pickup ploy to his older brother in 1942 when he was 7 years old; Micki O'Connor played it with his brothers and sisters in the 1960's and Chaz Larson was invited to play a game with his cousin in 1972. So if there is a connection with the novel, the novel must be named after the game rather than vice versa.