It depends on the type of plastic. Some plastics are very difficult to drill without cracking, even in the circumstances. When drilling something molded/cast, not flat, etc, it could be more tricky. Conventional twist bits are also part of the problem - they tend to auger in, and that contributes to cracking. Sometimes the best bit for plastic is made by cutting a nail or coat hanger and just drilling with the rough end. That'd probably be a bit small for what you're doing. Another option is grinding the holes with a dremel like tool.
Melting through the plastic is another option, though the fumes are very toxic.
Don't forget some gromets for the holes.