What is a chopper or a bobber? A chopper generally refers to a motorcycle that has had parts or the frame chopped and or modified in some way. A bobber is a style of bike that originated in the 1940's after World War II. They were created by simply removing any unnecessary parts, such as fenders, windshields, saddle bags, etc. These style of bikes have evolved greatly over the years. The chopper was the child of the bobber and now the styles tend to be interwoven.
People young and old love choppers, but having one built by a pro can be expensive. Obviously the cheaper option is to do it yourself by modifying a cheap bike. It seems that now the cheapest bikes are older Japanese models with some wear and tear on them. You will often see these bikes for a few hundred dollars, sometimes more and sometimes free. It all depends on condition, rarity, title, and the owners ability to part with it.
I would look for a bike that is a non collectors item, in good running condition with a clear title if at all possible. This will save a headache and money in most cases. You want to get the bike in good mechanical condition before the chopping begins. If you get a bike for cheap that needs a lot of work, it may cost you just as much to fix it as it would have been to just buy a motorcycle in good condition to begin with. Getting a title for a bike without one in some states can be tricky and difficult. I'd avoid the mess and just buy a motorcycle that has a good title. Look for something with some minor cosmetic damage. You were going to change it anyways, so why not try getting one with some minor damage. This can save some coin. Look for non collectors bikes, they cost more and if in good original condition you'll feel bad for cutting it up.
Building your chopper is all up to your imagination and skill level. The great thing about starting with a cheap bike is if you don't like your finished product, you didn't loose very much money. You will also learn how to work on bikes and be proud of what you did, instead of what you paid someone to do for you. If your not confident, start with simple items and go slow. The more you do the more you'll learn. The key thing to do is research and ask questions of those that have gone before you. Good luck and happy choppin.