Would love to advise you on your eBike build. Unlikely you'll spend the countless hours I have mulling over various possibilities, and there are lots of them, so lean on me. The magic formulae of a <$1000 eBike may exist, with compromise, but what do you compromise?
The things you need: 1> a worthy bicycle, 2> a hub motor kit with controller & controls, laced into a rim, 3> a matching battery and charger.
All <$1000 eBikes will compromise one or more of the above. I suggest not, so spending a bit more to get decent quality for all three. Major cost savings will come from ordering up a Walmart bike and doing all the assembly work yourself, including kit install and battery.
I liked the Walmart Avalon: http://www.walmart.com/ip/26-Avalon-Men-s-Cruiser-Bike/21635130
. It has the component set we've discussed: a dual suspension steel frame, Shimano 7-speed gear system, and simple caliper brakes. Price: $107.
You won't need more than 7 gears. Latter you can upgrade the front chain ring to a larger ring to match out your top speed pedalling at full throttle.
Get the EM3ev upgrade 8T motor kit without battery: http://em3ev.com/store/index.php?route=product/category&path=36
. Get it with the best Infineon controller - a 12fet 3077. You'll be pushing it with a 48 volt battery. Get it laced into an Alex rim. Price: in the range of $600 complete, shipped from China.
Get the rear motor with the 26" Alexrim DM26 CNC and Sapim spokes. Black spokes with a black motor, black rim & tire looks good. Get it with the HWBS sensor eBrake so it goes on the bikes existing brakes. Throttle choice is either, but I like the thumb throttles best.
Add in: the 7-speed DNP 11T freewheel, a DNP extractor tool, a Grin rear torque arm, 6 pairs of Anderson 45 amp powerpole connectors for the battery harness.
You'll need to get yourself a decent soldering gun & solder and 12 gauge wire in red & black to make up the battery harness, among other electrical chores. As well as some household tools if you don't have them. Pliers, screw drivers, hex wrenches, etc. A good vice is also nice to have. Amazon, eBay, Home Depot stuff.
The battery! Here's where a lot of kits and production eBikes compromise, because decent battery with sufficient capacity costs. I considered a lot of possibilities for you and settled on Headway: http://www.headway-headquarters.com/48v12ah-diy-headway-battery-kit/
. Its a well regarded LiFePO4 battery. 48 volts is what you'll need. 12 amp-hours is OK. You're saving by getting a kit & making it up. By doing so, you can also service it over time replacing out dead cells. Price: about $600.
Add in one of their 48V chargers. They have a 4amp, 10amp and 15amp. More amps means a faster charge, but also more cost. Your choice. 10 amp price: $154.
You're looking at about $1500 total, approximating shipping costs. Again, to make it less, you'll have to compromise and you'll end up regretting it or not using it. Idea: get your mom to give you some of this for your birthday.
In addition to that, you'll need to figure a method of attaching the battery pack and controller to your frame. I use a decent quality rear rack. Made up my own pannier to hold the battery.
Let me know what else you need from me. Happy to advise and help as I can. Look forward to my next visit in December and taking that eBike ride around Austin.