Not sure about cutting the plastic... as I have had issues with those too... you might try a spade bit, which scribes a hole, rather than the hole saw which tends to make a rougher cut.

Or perhaps a dremel tool and work out to the 3/4 hole.

It depends on the type of plastic... PVC type plastic (that gray stuff) tends to cut pretty well and not shatter. If the plastic is as brittle as you think it is, it may not be very UV resistant and eventually fail anyway.

But the better idea might be to use a metal watertight outdoor electrical box with a watertight cable connector... both available at your local Home Depot and probably more UV resistant than a project box. The other nice thing about these boxes is that you can get a front cover plate which you can drill easily as they lay flat... the cover plates have gaskets.

See the following links for examples of the boxes.
http://www.homedepot.com/Electrical-Electrical-Boxes-Conduit-Fittings/h_d1/N-5yc1vZ1xgcZbm4m/R-100211731/h_d2/ProductDisplay?langId=-1&storeId=10051&catalogId=10053
http://www.homedepot.com/Electrical-Electrical-Boxes-Conduit-Fittings/h_d1/N-5yc1vZ1xgcZbm4m/R-100404096/h_d2/ProductDisplay?langId=-1&storeId=10051&catalogId=10053

Cover
http://www.homedepot.com/Electrical-Electrical-Boxes-Conduit-Fittings/h_d1/N-5yc1vZ1xgcZbm4m/R-100207344/h_d2/ProductDisplay?langId=-1&storeId=10051&catalogId=10053