You will need to get hold of an old mower. Remove all the parts till you have only the chassis portion remaining. You will also need to sandblast it and clean it to remove dust and rust on the surface.
The next stage will be to strengthen the chassis portion; you could use metal to do this. You can also use metal for parts located on the back axle. Either get some professional help in welding or try yourself - but only if you’re experienced.
Get yourself a six speed heavy duty rear end. Tear it down, Inspect the gears, and send them in and have them heat treated and hardened. Then put them back in and fill it with a very very thin oil like 5W/ 20.
You will next need to cut out the engine plate. For this you can use the jigsaw, a drill, and a hole saw. Then you will need to procure a steering set, wheels, engine, and some spare tires. You can easily get cheap spare parts by looking in the classifieds section of papers and magazines. If you prefer buying online you will find plenty of sellers willing to provide good quality spare parts. Most of them supply spare parts for go-karts.
Now you need to bolt and nut or weld the steering set to the chassis. Be sure to wear protective goggles if you weld, as the sparks can be blinding. Next you will need to fit the wheels, rear transaxle, and tires. If you’re new to this you can get help from a professional. Afterwards you will need to fit the brake calliper as well as making your own pedals to accommodate the clutch, brake and accelerator. Then go on to fit the engine, clutch, belts and pulleys. For the steering you could either weld a frame for it by yourself or get help from a professional welder. You will then need to procure a steering wheel and a seat. Try and look in your local junkyards - you’ll be sure to find the type you want!
Exhaust Set Up
Many people like to go with the thick mild steel. I personaly do not think that that is the greatest idea, I think that a thin walled Conduit would be the best.
So here is what you do, you get yourself 10' of 1 1/4" electrican Conduit and about 5" of 1" mild steel. you take the mild steel to your local hardware store and have it threded to fit your exhaust port. Than you bend the other pipe to yuor desired liking. It works and sounds best when it is at least three feet long. Finaly clamp it on.
Once you have finished making the lawn mower racer you can take it for a test drive. This will enable you to get a better idea of how it runs, as well as locating problems in the steering or brakes. Try riding on different terrains like grass and bitumen for a better feel.