Be careful and remember there are dangers that pepper spray can't handle.
It takes a lot of research of preparation, but this is an idea of it:
If you plan on camping your way across you'll need food (in a soft cooler), a stove, cooking gear, eating utensils. Sleeping bag, tent, tarp, sleeping pad. Rope is a good idea (tie the tarp up in the rain), fire stuff, insect stuff, warm clothes, rain gear.
Then you need bike parts like chain, tubes, derailleur cable, brake cable if applicable, bike tools, grease). You'll want a trailer and panniers and whatever other bags you need to hold all the stuff.
It's a good idea to bring along a flashlight, extra batteries, an extra battery for your cell phone, a first aid kit, a camelbak water pack, water bottle and cage.
There are a million and three routes you can take. If you've got good senses of direction and adventure you can wing it, but otherwise route planning is important and something to do with consideration of what's best for you, as a bike. Make sure you always have a place to restock your supplies and a place to sleep at night. Get maps with the motels and campgrounds on it (AAA has them) and use them with the road maps to plan your way.
You'll want cash on you at all times, plus a few thousand to spend on your way.
And don't forget your camera.
People have done this before:
http://bikeacrossamerica.org/
http://www.bikingacrossamerica.com/