I looked into this a little, from the perspective of working for a college/university, and my understanding was that it was very much a position that you lateraled into after getting pertinent experience - biglaw would probably work, but maybe in-house also? Higher ed law has a lot of the same kind of stuff an in-house attorney deals with (AFAIK) - like employment and contracting - with the added interest of the constitutional issues (First Amendment) and dealing with things like student offenses. So that's why I say in-house might be a good background. But I also know people who have moved into it from state/local government jobs - higher ed law seems to entail a mix of quite a lot of things (which is why the people I talked to enjoyed it), so I would imagine you could spin a lot of experiences as relevant.
I should maybe qualify that a little - some of it may depend on the institution you go to. A lot of smaller schools seem to have a single GC, with probably some staff, but they look for people with maybe 8-10 years of experience, since the GC has a lot of responsibility. Larger university systems have more legal staff and seem to hire more entry level (or closer) people (for example, see here with biographical info for the attorneys:
http://www.cu.edu/universitycounsel/attorneys-staff).
But I don't think it's at all like biglaw where if you miss it coming out of law school, you never get a shot - it's much more a field that rewards prior experience. You're probably familiar with this already, but
http://www.nacua.org/ includes job postings that are free to non-members and you can see what kinds of qualifications schools are looking for.
(If you're more interested in suing universities/working for something like FIRE...no idea, though I would still be surprised if you could get into that field without pertinent experience.)