The closer to the ground the camera is, the faster the speed will seem.
Unfortunately, the closer to the ground the camera is, the more likely that it's outside the car and therefore stands a very good chance of getting destroyed, and you can't afford to be replacing cameras all the time like the network guys can
If I were doing this, I'd mount the camera behind and slightly to the right of the driver's seat. Get an over-the-shoulder shot of the driver, make sure you can see one (ideally both) of his hands on the wheel, and a good shot of outside. You may need to set it on manual iris if it has one so that it won't overexpose the outside in order to expose your hands. You should be going for underexposed you in order to get a properly exposed racetrack.
If you have a shock mount or your image stabilizer on your cam is good, zoom in a little bit. The wider you are, the slower it looks like you're going. Don't zoom in too much, though, or it'll be so shaky that it won't look very good at all.
See if your friend has a cam you can borrow and mount it under the dash, pointing up at your face. try to get your legs in the shot.
Then you can edit it together, sequencing the cornering with shots of you downshifting and turning. You can make it appear faster this way since the changing of camera angles will hide the fact that the race footage looks slow.