You can build a lot of bike for $1K in parts but for simple and reliable you can do a build like this for ~$600.
49cc HS 142F 4-stroke engine and Grubee 4G transfer case ~$350 at http://gasbike.net
29" Macargi Fatal Love steel frame beach cruiser ~$240.
I get mine local but they can be found on-line.
Note you will need to put a front brake on it.
It's a little slow off the line but winds out over 30MPH, legal most anywhere, and pretty easy to build.
The thing is allot depends on your stature and travel needs...
If you are smallish like under 160 pounds and less than a 28" inseam that big bike will be too much bike for you, go with a little 2-stroke and smallish bike like this 48 or 66cc 2-stroke on a Macargi Touch for like $350.
Options abound everywhere in between for bike and engine choice including making an electric shifter which is one of my favorite ways to build but not cheap due to good battery choices.
Depending on the bike price you can do a fairly small battery electric build like this for under $1000.
If memory serves I paid $300 for the Caddy NuVinci shifter bike and $600 for electric system.
Or of course you can go BIG power like this 7-speed electric shifter build I just finished yesterday that will flat blow you away for ~$1000.
Gears change everything in drive train, especially on an electric but adds a hundred or two, not very easy to get right and there is more maintenance involved but no kidding that electric goes from 0 to 20 in like 2 seconds from a dead stop in 3rd gear and when you rev out the motor you just jump to 6th and can hit 35MPH in about 6 seconds.
No clue what this new build tops out at but like I said, it blew us away ;-}
I suggest you just go to a REAL bicycle store and test ride some bikes with a nice center cavity to find what fits you best and you enjoy just pedaling, then pick what power plant and drive to use that will fit it because well, if you are not comfortable on the bike just pedaling you'll hate it with motor power.