Hi bens89,
Working in a family-owned business may be different from the traditional experience of working in large, recognized names, but it’s not a disadvantage. In fact, helping a struggling business and turning it around (if that happened) can be a solid, in-the-trenches, diverse experience that lot of B-schools value.
The way you convey your experience won’t be fundamentally different from what ‘more traditional’ applicants do. You do this through your essays, recommendations, and resume. Essays are less about achievements and more about the stories, effort, lessons drawn, your working style, and other softer parts behind them, which leads the reviewer to draw positive inferences about your managerial abilities. (Just mentioning ‘I convinced the client to my proposal’ won’t be enough.) It’s also important to show understanding of nuances when making leap-of-faith assertions (‘I want to start a company’ generally won’t be enough) or projecting ‘fit’ with a school. Also, if you’ve any weak areas in your profile, you should address them in the optional essay.
Recommendations provide a third-party validation of some of the things you’ve mentioned in your application. Your recommenders should ideally be persons who know you well, and, very critical, can back their statements about your candidacy with insightful examples (it’s better to go deep in one experience than provide smattering of 2-3). They should ideally support some of your strengths and allay concerns on your weak points.