Lane’s purple hair is very obviously a cheap wig – it doesn’t fit properly and looks a different style to her real hair. The whole hair dyeing plot in this episode is not possible in real life. You can’t bleach black hair blonde, then dye it purple all in the matter of a couple of hours (it would take days). It also isn’t possible that Mrs Kim wouldn’t have smelled the bleach and hair dye while working in the antiques store on the ground floor.
However, none of this really matters much, as after a very short celebration of her bold new look, Lane has a crisis of confidence, and insists that Rory dye her hair black, so that her mother won’t find out what she did. This is, of course, only adds to the implausibility of the episode.
The Annotated Gilmore Girls 