This is the song playing at Lane’s house when Rory is bleaching her hair, so that it can later be dyed purple.
“Cities in Dust” is a 1985 song by English rock band Siouxsie and the Banshees, released as the lead single from their 1986 album Tinderbox. It’s a goth rock and dance-pop song describing the 79 AD destruction of the city of Pompeii by the the volcano Vesuvius. It went to #21 in the UK, and #17 on the Dance Club charts in the US, the band’s first significant success.
The song has quite often been chosen for soundtracks, including by The Marvelous Mrs. Maisel, another Amy Sherman-Palladino TV show.
The lyrics about the burning down of the city seem to mirror Lane’s feelings that she is burning all her bridges by dyeing her hair. This is reinforced by the fact that the bleach soon starts burning her scalp.