I have not let my SM066 get to that point, but I think you should first start with a non-scratch hard plastic paint scraper and get off whatever stuff you can from the smoker box. I guess then I would maybe heat up the smoker (only to 100F or so) and try a damp/wet coarse rag (like terry cloth) to see what else you can get off. I would not use any oven cleaner-like product, but some grill makers, like Weber, sell a cleaning compound that doesn't cause your cooker to retain those noxious chemicals.
If your cooking grates are chrome rather than stainless, your scrubbing options are limited to preserve the plating. If they are really bad, you might want to spring for the SS grates. They are pricey, but you'll never have to worry about the surface finish again.
CS customer service may have other advice, they are really good. If you're new to smoking, understand that the smoker will never be really "clean" - it will have a burned-in layer of smoke and grease that will actually enhance your product, and can be minimized when necessary by wiping with a damp rag after each smoke. I myself clean my cooking grates after every smoke, but that's just me.
Good luck, and please report back on what does/doesn't work.