I am currently reading the Tsutsumi Chunagon Monogatari and the Konjaku Monogatari Shu and am absolutely fascinated. Outside of the classic literature produced at the time, what are the books to read to better understand the Heian period ?
Ivan Morris' The World of the Shining Prince is what is usually considered the most comprehensive single volume on Heian Japan, particularly the middle of the period when Murasaki Shikibu lived and wrote.
Royall Tyler's Disaster of the Third Princess is an interesting analysis of several often-debated issues in The Tale of Genji. Apparently Ukifune didn't try to drown herself?!
I recently finished Fumiko Enchi's A Tale of False Fortunes. It's more historical fiction, and a meta on writing/narrative itself. Its plot revolves around the power struggle between Michinaga and his brother Michitaka's children after Michitaka died.
I haven't read the following myself, but you can:
Francine Herail's Emperor and Aristocracy in Heian Japan; Haruo Shirane's Bridge of Dreams: The Poetics of the Tale of Genji, and Envisioning The Tale of Genji.
Folks have already mentioned Genji, which is certainly a noteworthy piece of literature. I'll therefore suggest the Pillow Book by Sei Shōnagon, a book of observations and musings recorded by Sei Shōnagon during her time as court lady in the late 10th century
You may enjoy reading these http://digital.library.upenn.edu/women/omori/court