What it will cost you depends on your ability to make stuff, the tools you have access to (own, rent, borrow, etc) as well as what materials you can get your hands on cheaply.
Some start with a "dead" kart and fix it up on the cheap, others pull engines out of junkyards or off lawn equipment that's been tossed to the curb, others trade and barter their way to a complete set of parts and build from there.
The go-kart I'm building for my son started off as a massive piece of discarded exercise equipment - about 200 lbs of steel tubing both square and rectangular, so this had no cost other than the gas to haul it home in my truck.
To make the other parts, I've spent about $200 plus $30 for bearings and maybe another $50 in hardware (heim joints, bolts, nuts, washers, i.e. the little stuff). The Audi differential I'm using was very cheap - $35 plus shipping, and the electric motor (4.5 HP) cost me $110 shipped to the door.
What's going to cost me dearly is the controller... which is why I put the effort into designing my own and soldered up a "tiny" version of it to see how it might actually work. Once testing and all the changes are finalized, I'll be making a 300A version to drive this electric motor. The test version can only run a 5A motor, just enough to test functionality and find mistakes. Blowing a 5A fuse is safer than blowing a 300A fuse
Anyway, here's my kart:
http://www.diygokarts.com/vb/showthread.php?t=4634
Be warned, it's not for those who are shy.
