You're lucky your school gives you plenty of access to equipment. But even so... I feel it is important to have a camera of your *own*. As when you own one you get to know it so much deeper, than one that you just rent or book out. So I'd suggest you pick up perhaps a cheap Panasonic GH1, they are going second hand on eBay for under US$200! *NOTHING* in this price range can come close to matching what the hacked GH1 can do! (heck, I have used both a GH1 and a 5DmkII and I much prefer the GH1 for filming with! And one of those is waaay more expensive than the other...) This way you then have a very cheap camera of your own to use throughout filmschool. And only after you graduate, go splurge out on a nicer camera. How long is your course? We might be up to the Panasonic GH6 by the time you graduate!
The other alternative, is go for a camera which is a bit of a midway point in price and performance between a GH4 and a GH1. The top choices would be: Nikon D5200, Sony A6000, or Panasonic G6 (a new G6 is in my opinion a much better value option than a used GH2 which goes for about the same price). One of these would be quite nice to have over a GH1, but are less important than simply *having a camera*. Any camera! Even an old Panasonic GF2/GF3 will do (that you can find for as little as fifty bucks second hand if you hunt for it)
Because having a camera always at arm's reach which you can pick up and use at a moment's notice is so much more handy than your school's equipment, which even if you live on campus will still mean a half to the faculty, filling out forms etc... before you can shoot it. And the easier you make it so you can just get up and go shooting, the more you'll do of it! And the more you do, the better you'll become. Practise makes perfect! So while you don't have to blow every cent on getting the best camera body that you can, because you've got the filmschool equipment available, I still feel you should get "some sort of camera".
Then once you have your camera (and a basic lens or three, you can pick up set of 3 c mount primes for it brand new for less than a hundred bucks!), either don't spend a dime more and put the rest in an Index Fund (or whatever is your prefered place to keep savings) until you graduate. *Or* you could spend the rest on equipment, but in areas that hold their value longer. Thus once you graduate, your equipment will still be a big asset for you. A tripod is a good place to start. The technology around tripods is fairly set in stone, and will not change or advance too much over the next five or ten years, or even longer. This guy gives great advice for students and others on a budget: https://www.youtube.com/user/NitsanPictures/search?query=tripods Personally I got a Fancier FC-270A (and the taller version with longer legs), which I really like. And also got the Beike BK-777 as well, because it is soooo much lighter!
You might think lighting is another good area to invest in too, but I wouldn't be so sure, with how very very quickly moving LED and battery technology is going forward. It could be a quite different world in another 5, 10, or 20 years from now, than what it was the same time ago in the past. You've got a total budget of 5 grand, after buying a GH1 (or even a G6) that still leaves an awful lot on the budget. I'd perhaps (if it wasn't going into an Index Fund...) just be putting almost token amounts in each category just to get them covered. Then put the remaining half in that Index Fund! So for instance: Panasonic GH1/G6 + lenses + tripod + reflector + 3x Yongnuo YN-300 LED lights (with batteries) + Tascam DR-60D + Rode NTG-2 (depends on your approach to sound though... you could just get a SG-108 for thirty bucks if you want to be very frugal, and a Tascam DR-05 or Zoom H1 recorder). Now depending on how much you spend on lenses (anywhere from a hundred bucks, to several thousand), this set up could cost anywhere from comfortable under your budget to only a tiny tiny fraction of your original budget.
For lenses, I recommend the Nikon F mount (as they have the most versitile use in their ease of adaption to other mount, so they'll be fairly "future proofed" in that you can take them from one system to another) unless you go with the 3 C mount lenses which I suggested beforehand (on eBay: 25mm f/1.4, 35mm f/1.7, and 50mm f/1.4 for a total cost brand new of about a hundred dollars. This is what I went with when I first got my GH1). The good thing is lenses depreciate far slower than camera bodies, so they'll hold their value well (if you buy second hand you may even see their value appreciate over time!). A suggested list would be:
Tokina 11-16mm f/2.8
Sigma 18-35mm f/1.8 (or the cheaper Tamron 17-50mm f/2.8)
Nikon 50mm f/1.8D
Rokinon 85mm f/1.4
Together with a standard adapter and an RJ Lens Turbo, this would have you very well covered for a wide range of needs.
http://www.personal-view.com/talks/discussion/9086/rj-lens-turbo-m43-adapters/p1
These gear suggestions I've given you'll still find useful after graduation no matter what, even if there is a flashy "Panasonic GH8" or you are shooting with a "Sony F8" (some made up sucessor to the Sony F5) or who knows what! As they'll always find a place somewhere, be it as back up equipment, or loan equipment for friends, or as a "crash cam", or to take when on holiday (or any other risky situation, not just the beach!). And if you do sell them a few years from now, their prices are such you shouldn't lose anything much on the deal.