"Backpacky" is not a standard English word, but it would describe someone who is likely to use a backpack. Such people are generally young, probably students, and probably a sort of a hippy -- or at least tries to be like a hippy.
Grunge, of course, is grunge rock, a particular style of music. Kurt Cobain was the front man of the grunge rock band Nirvana, and he died some years ago.
If you say somebody is "into" something, you mean they are very interested and active in it. Presumably Claire is asking if Lisa is the sort of person who would listen to grunge a lot, go to concerts and perhaps dress like someone from Nirvana and have posters of Kurt Cobain on her bedroom wall. In this context, "way" is an informal way of saying "very".
Student or Learner