You're asking the million dollar question... how do you really get the attention of a major record company and make them listen to your demo?
The answer is 'by any means necessary' as there are absolutely no rules for this one.
You need to identify who at a company it would be most beneficial to get your product to. It's usually the A&R personel but record company receptionists will give you this info freely.
Try to set up meetings or accidentally meet them out and about.
Persistency pays in this area, although there's a fine line between being persistent and being a nuisance. Dont shoot yourself in the foot. It's all about patience.