The hard copy letter would most likely be neither here nor there for most CPs - e-mail is fine and more easily organised for recall.
As advocated several times above, the best pitch is an actual visit. Most experienced pilots know the pain of this, but unfortunately our industry has shaped that way and although I go against the notion of "incompetent CPs" raised above as a reasoning for their lack of response, I would offer that busy CPs have lots of things they need to be doing and the best way to get their attention is to actually be there physically. A decent CP will take the time to say "g'day" or alternatively will have some "local procedure" for when it is best to call in. Remember, the CP's job is difficult enough without being required to constantly respond to solicitations from hopeful pilots, especially if the Company is not recruiting and has not advertised. It is nice when a CP does respond, though, but the sheer volume of applicants would preclude a response to everyone and although that may be seen as rude by the applicant, bear in mind there is two sides to every story. So, stand out from the crowd, but be patient and realise the greater the effort, the more likely your success will be for at least getting an interview!
Never buy a CP a beer to try and win a position...
If you have questions of a CP that you feel need to be discussed before physically visiting, maybe you should try an e-mail to the CP with those questions and maybe he / she will respond in detail at their convenience. Where a CP is currently busy when you call (say, doing a flight check or similar if they are an ATO) they are not likely to drop everything to take your call. This is not a lack of professional courtesy! I'm sure you'd be a tad annoyed if, during your check with the CP, he interrupted you to take a call from a new CPL interstate who he / she has never met hoping to secure a job...