Okay, for an electric kart to meet those goals, it's going to take
A) More than $550
B) Much more experience
I consider myself a somewhat seasoned electric kart builder. To achieve those speeds, you're going to need loooots of current, which means looooots of expensive batteries, which means an expensive controller to handle that juice. Not to mention the extreme danger that comes with that much power, you could seriously hurt or even kill yourself, which is why I would suggest lowering your standards. Second, your run time is unrealistic. Even electric cars which have large advanced technology batteries can barely go for that long. (Tesla:~2hours, Volt:~45min-1hour) I would shoot for more like a 25-40min. depending on how hard you drive it/riding conditions.
So, for a first time build:
What I do, really easy, cheap, can be done easy in your budget.
1. Go on craigslist, find a high wattage electric scooter (something greater than 300watts). This will get you motor, throttle, controller, chain, sprocket, switches, fuses, sometimes brakes. Most people will sell their scooters because they can no longer keep a charge, 95% of the time, new batteries will fix that. You can usually (in my area) get a scooter like I described for >$150.
2. This scooter will probably need new batteries (~$50-$100 depending on the number of batteries i.e. 24v 36v 48v, and what run time you are looking for.)
3. Design/Build frame. (~$100-whatever you want to spend)
4. Apply electronics
5. Have fun on your new toy.
I can give you more specific information if you tell us more about what you are looking for as far as what kind of go kart you are looking for. (big, small, weird, ect.)
These are my electric builds I have done using essentially this process:
Ride-On-Cooler:
http://www.diygokarts.com/vb/showthread.php?t=11489
Electric Radio Flyer Wagon:
http://www.diygokarts.com/vb/showthread.php?t=13491
Beer Crate:
http://www.diygokarts.com/vb/showthread.php?t=13523
