Josef Chuchla started for Tělovýchovná Jednota Vesko Nová Ves during his playing days . He contested his first race in the Skeleton World Cup in Altenberg in 1995 , where he finished 29th. In the same year he took part in the 1995 Skeleton World Championships in Lillehammer and reached 30th place. After he competed steadily but without outstanding results in the 1996 World Cup, the Czech did not have any international appearances in 1997, it was not until 1998 that he appeared again several times. In February 1999 Chuchla drove on the combined ice rink at Königssee as 25th on his best World Cup result. A little later he finished 23rd at the 1999 Skeleton World Championship in Altenberg. In 2001 he won a Skeleton Challenge Cup race in Calgary and placed very well in several races of the new European Skeleton Cup . In the overall ranking of the European Cup in the 2000/01 season Chuchla was fourth. At the 2001 Skeleton World Championship in Calgary, he finished 21st.
Since 2001, the World Cup appearances for Chuchla have become less frequent, but he continued to compete in major events. The highlight of his career was his participation in the 2002 Winter Olympics in Salt Lake City . It was the first time the sport was held at the Olympic Games since 1948. The Czech finished 24th. At the 2003 European Skeleton Championships , he reached 20th place, and in 2004 he was Czech champion.