Start with a personal email. Keep is short. Your ask, in the initial email, is for a phone call. That's it. Your objective is to set a time for a call. One sentence on what you're building, and one sentence on why it is applicable to them. Indicate that you respect their opinion and that you would like them to help shape your product roadmap.

On the call. Don't sell them anything. Ask them questions about what they are currently doing, and what solutions they would like to see for the "problem." Even if you just talk to 1 out of 20, listen to what they want and tweak your product to make them happy. If that customer is happy, then there is a good chance that other users will want what you've built. "Do things that don't scale" - PG.

Your goal on the call is to get that person using your product. Offer to do it for them. Write code for them. Help them implement your solution. Treat it like a consulting gig. Whatever they are looking for, you can do it.

A LinkedIn add compliments a direct email. But, a LinkedIn message is often ignored.

If you don't have the customer's email use Rapportive -- http://rapportive.com/

Rapportive is amazing! You add Rapportive to gmail. It is a free add-on for Firefox, Safari, Mailplane, and Chrome. You can guess at the email you are looking for...firstname@companyname.com, firstname.lastname@companyname.com, firstinitiallastname@companyname.com, etc. When you get a hit the right hand side of gmail lights up with the person's picture and links to their publicly available social networks, all of which gets displayed alongside the message. The sidebar replaces Google’s contextual ads, so that’s a double win.

Even if you don't like my advice you should checkout Rapportive, it helps.