The work is an episodic biographical saga following the life of "the shining Genji", a courtier and public official who is an inveterate womaniser. The backdrop is an aristocratic world dominated by intrigue between the rival families of the consorts and mothers of emperors current, retired and deceased, a world of which Murasaki, a court lady-in-waiting herself, had first-hand knowledge. (Actual "affairs of state" are never elaborated on, however, while the peasant rice-farmers who supported the Heian state, and provided the "emoluments" on which its aristocrats depended, are not mentioned at all.)
The obvious interest is in the extensive cast of characters and their interactions. These are not always that appealing, but their dilemmas and concerns are fairly universal in their appeal, despite their location within a fairly rigid social system, and their individuality comes out clearly. For the modern reader, The Tale of Genji also offers an internal view of a fascinatingly different milieu. One notable strangeness is the emphasis on aesthetics, with courtship by exchange of poems and a prominent role for smells and flowers.
I picked the 1976 Seidensticker translation over the 2002 Tyler translation because reviews suggested it was less strictly accurate but more readable. And, though I bought the full work in the attractive Everyman hardcover edition, I chose to read the 1990 abridgement published by Vintage, borrowed from a friend. This is actually more a selection than an abridgement, consisting of 12 of the first 17 chapters, about a third of the work, with almost no modification. (To the extent that Seidensticker prefers to add footnotes such as "This paragraph and the succeeding three paragraphs refer to a scene in a chapter which has been omitted".) This edition also includes a twelve page introduction, discussing the new translation and its relationship to earlier ones, notably Arthur Waley's, and explaining the abridgement; it also has some woodcuts from a 17th century edition.
Based on this sampling, The Tale of Genji is enjoyable and accessible enough that I hope to find the time one day to read the remaining chapters.