A rat rod is a throw back to the old way hot rods were created. Today the term hot rod generally refers to car from a specific era designed for speed and racing and not much else. A rat rod is the same idea but without any preconceived notions of what it should look like. When you build your own rat rod you guarantee you will have a vehicle unique to you.
Pick out your engine; this will depend a lot on what kind of frame you are considering and how much weight you need the engine to power. If you have a big budget you could shop around for an engine online and pick it based on performance specs. If your cash isn't as plentiful, the best option is to go to your local junk yard and see what you can pick up for a great price.
Build or acquire the frame. Since this is a rat rod you can use whatever you want for the frame as long as it is strong enough to not fall apart under the influence of the speed you will hopefully be achieving. Light weight metals and composites are the trends right now for racing machines; however, cost will again be the biggest consideration.
Mount the engine in the frame and install the basic power components. These will include things like the transmission, battery, wheels and axles. If you are starting your rat rod with a junkyard car you should be able to find parts that fit together easily for your basic structure. As far as electrical systems go, the addition of things like lights should be afterthoughts more than any thing else, as a true rat rod is not necessarily street legal.
Test the heart and guts of the car; this means making sure the engine will start, run and turn the wheels. As long as you can get that accomplished the rest of the rat rod will be easy.
Install the steering control mechanism and gas and brake pedals. The placement of these are up to you, however make sure there is room for some sort of seat, no matter how crude, that can reach all of the necessary controls.
Finish up the body and interior. As far as the interior, you can make it as elaborate or as simple as you wish, but anything you add for comfort also adds to the overall weight. As far as body panels, they should serve a purpose: increasing the car's aerodynamics or decreasing drag, or, like a spoiler, increasing the amount of tire grip. Any body panels you add will add weight, and contrary to popular opinion, painting flames on them does not increase your vehicle's overall speed.
Traveling at fast speeds without a proper roll cage is very dangerous, therefore it is recommended that you wear special safety equipment to minimize your risk of injury.