I wish there would've been people doing stuff like this when I was in school. Best of luck.
An interesting place to start would be Damien's £1000 build, including the car:
http://www.diyelectriccar.com/forums/showthread.php/1-000-euro-ev-build-193217.html
I'm doing a motorbike conversion, which is about as dead-simple as it goes, and I've broken my build progress down into each stage:
http://www.diyelectriccar.com/forums/showthread.php?p=993802
I presume you're not building the vehicle itself, you're doing a conversion?
Here's a bit of runthrough:
Vehicle: Pick something light, and simple. An old mid-90s civic hatchback or something like that. Ideally, something that didn't have power steering or power breaks. Mid-90s is a good era because things were made light weight, but hadn't been bogged down with sensors and electronics and other crap. You probably want it to be manual, but you can make either work.
Motor: You will be using an old electric forklift drive motor. For your budget, this is the only option. And it'll perform just fine. Bottomless torque. You'll get a 150-200lb motor that's 9-11 inches in diameter. Ignore the wattage/horsepower rating, it can handle much more than that. Don't email, don't call, just show up at a forklift salvage/service/repair place, ask if they have some time to talk to you, explain your story and ask if they have any motors, controllers, cables, fuses, contactors, etc you can have, or, if they have or will be getting some vehicles that you could use your own tools to salvage from some day. If you email it will get ignored, go in person, be kind and polite. It may take weeks or months for the right one to show up, so get started on this early. Most of your build will center around the motor. Take whatever the motor mates to as well, it will save you on machining which is the most expensive part. The motor should be free or nearly free, under $1/lb for sure. Generally not worth their time to extract but this varies based on the poverty of the city you live in. This motor will be a series-wound DC motor. It will not do regenerative braking, which doesn't matter, because it has nearly zero impact on your range anyway. Alternatively, you can try buying a junk forlift but note that you'll have to find a way to move a 5-10 ton vehicle that probalbly has dead batteries.
Transmission: Your motor can spin the driveshaft fast enough directly power the driveshaft. You don't need a transmission. Sometimes it's nice so you have a manual disconnect (neutral), sometimes it's nice for hill climbing if you're underpowered. But if you want you can get rid of one. Different challenges either way, but you probably won't ever shift gears, you'll just leave it in 3rd all the time, including starts.
Batteries: This will be your biggest expense. Don't buy lead-acids, they cost more for how long they last than the gasoline you'd be burning. Figure out your range. A small light car will use 350 watt-hours per mile at highway speeds, maybe 200 watt-hours per mile at city speeds. Decide how far you need to go, this will decide how much you have to spend or salvage. There's great deals to be had buying OEM salvaged vehicle packs, maybe, possibly within your budget. If you want to build your own lithium pack, go to every single computer recycler and laptop repair place in your city with some small rubbermaids with your name, phone number, and email address taped onto them, and a "Call when full." Offer to leave the tub there and tell them you'll take them away for a school project. Some might ask you to pay. $1/lb is a normal rate. You'll be spending at least $50 on a charge tester and months scrapping and testing cells. To get 60 miles of range will probably take you a year of saving up and testing, so, start early.
Speed/Motor Controller: Hopefully you could get something out of the largest forklift you could find. Buying used is at least $300, if you want a great one you can build yourself, probably $500-$1000 for a Paul & Sabrina Controller, but if you hunt Ebay for deals on the expensive parts you can get it for less.
...
That's the stuff you gotta start hunting for ASAP, because it shapes the rest of your build. The more time you give yourself to find deals, the more will appear. The less time you have, the more you will have to spend.
Best of luck.