His work spans multiple formats—novels, plays, librettos for operas and ballets, poetry, and essays on film and literature.
He has written several works on the Italian director and writer Pier Paolo Pasolini, including Wir sind alle in Gefahr, for which Klimke won the 1995 Ernst Barlach Prize.
His novella Der Test oder: Chronik einer veruntreuten Seele was one of the key works in German AIDS literature of the 1990s.
He received his secondary education at Freiherr-vom-Stein-Gymnasium [de] where his father Wolfgang taught German and English literature for many years and was editor of Thomas Wolfe.
At first he wrote for newspapers and worked as a translator from Italian to German for the publishing company Rowohlt Verlag.