Klimsch was born on 10 February 1870 in Frankfurt am Main to a family of artists, studying at the Royal College for the Academic Fine Arts in Berlin, and was then a student of Fritz Schaper.

[1] After the seizure of power by the Nazis, official commissions such as busts of Adolf Hitler, Erich Ludendorff and Wilhelm Frick, among others, were predominant.

After Nazi Germany's defeat in World War II, Klimsch and his family settled in Salzburg, Austria, but in 1946 was deported by the local burgermeister, Richard Hildmann, for being a German citizen.

However, shortly before his death in 1960, Klimsch received the Federal Cross of Merit from Hans Filbinger, the Minister President of Baden-Württemberg, on his 90th birthday.

Kilmsch died on 30 March 1960 in Freiburg im Breisgau, and was buried in Saig bei Lenzkirch, where he was an honorary citizen.