You will have to put in some serious time and energy to make these happen. Yet at the end, you will know the organization, be ready for interview questions, and build trust and credibility.

1- Know the organization inside and out.

Become a serious expert. My goal: Know as much as or more than my interviewers.
        
What would I want to know about the organization?
What the culture is like,
What the people care about,
How they work together as a team,
What makes the organization unique,
What kind of results they deliver.
What are their strengths,
What are their weaknesses.

There's also the basics like: how many people they serve, how many employees they have, how much revenue they make (or how big a budget for a non-profit), what is their reputation.
And there's the me part of it: What are the reasons I would be excited about working there, and what are the ways that I can help them be successful?
        
Where would I get this information?
        
First, I would read everything I could. Their website. Their social media pages (Twitter, Facebook, Google+, LinkedIn). Their press releases. Their newsletters. Whatever comes up on Google News and Google Scholar with their name.
        
Then, it's time to move on to an even more important channel, talking to people.
        
Talk to employees
        
First, I would want to talk with the people who worked there or used to work there. They can tell me the good, the bad, and the ugly, and give me a ton of fodder for why I would want to join the team.
        
How would I find these people?

One easy way is to use online social media like LinkedIn or even Facebook and search for folks who worked there. When I find someone, I would do everything that I could to connect, even if I only know someone who knows someone who can give me an introduction.
        
Yet, if that doesn't work out, and I really want the job, I won't stop there. I'll reach out to my friends and family and anyone I know to find connections (past employees are great too). Shucks, if I got stuck, I could even post $100 reward on Facebook to my friends and family if they can make an introduction.

I'll look at my university's alumni database or if there wasn't one, I'd call my school alumni office.
        
I'll reach out to professional associations to see if they have any connections for me as well. I'd even ask members of groups that I am a part who they knew (volunteer or religious groups, sports leagues, etc).
        
When I did connect with these folks, I'd ask them lots of questions, ranging from the best part of working there to what questions they faced in their interview.
        
Talk to customers

If I were a teacher, I'd want to talk to students and their parents to learn about their experience with a school. Patients for a medical practice. Customers at a store. And I'd ask them about what their experience was, as many details as I could.
I'd use similar tactics to those I tried above to find them.
              
Talk to other stakeholders...

Who else knows the organization? The board members at a non-profit, a salesperson who provides them with services. The medical students who rotated through the hospital floor. Someone who works for a competitor. Maybe there are others in your field.

By the time I am done...

I want to know exactly why I would enjoy the work, what projects I would contribute to (or would volunteer to be a part of), and ideas to make the organization better. I would know how their organization's mission fits with my personal goals and what I care about.
        
2 - Study the field.

I would want to make sure I knew the business that they were in.
        
The principles and the science behind what they do. The new trends coming out -
        
Whether that's understanding a school district's new policy on standardized testing, reading about the newest dental picks, or reviewing the financial concepts behind a hedge fund's trading strategies.
        
My goal for all of this:
        
I want to have the ability to walk into an interview with the President/ /Director/Chairman of the board or whatever bigwig they threw my way and be able to hold my own.
        
3- Build trust and credibility

I want the people in this organization who may not know me to trust and believe in me. Which isn't always easy to do.
        
Yes the interview is a big part of this. Yet, there are other ways to build my case.
        
For all of the people that I spoke with to learn about the organization, if I knew any of them even reasonably well, I would ask them to give me a recommendation. Make a phone call, write a note, send an email. Without the organization asking, just send something over that says they believe in me and know I am passionate about this job.

Of course, I'd speak with my references too. Tell them exactly why I am excited about the job, help them identify my accomplishments that are relevant to this position. Make sure that when they are asked, they can deliver a 10 out of 10 recommendation.
        
Depending on my field, I'd love to have a portfolio to present, examples of my past work that are further evidence of my value.
        
4 - Prepare thoroughly for my interview
        
I wouldn't just wait until the day before the interview to start preparing. I'd brainstorm a list of critical questions, and write my answers to them. Then rewrite them and use whatever tools I could to make sure my answers were as strong as possible.

Then practice a lot. Friends and family could give me mock interviews and provide me with feedback on my answers. By the end of it, I'd know what I would want to say inside and out.
        
I'd research my interviewers too. Get to know who they are, what they care about, and any areas of commonality between us. So we could quickly fall into meaningful and enjoyable conversation.
        
5 - Follow up with a personal touch
        
Send each person an individual thank you note. If we discussed any topic that seemed of special interest, I'd dig up an article that would continue the conversation and show I paid attention.
        
And if I didn't hear back within a week or two, I'd follow up, just to remind them of my interest and get some peace of mind, not to try to sell myself.

And no matter what happens, I'd reconnect with every person at that organization, for a quick call or cup of coffee if possible. I'd thank them again for their time and for our conversation. We could even discuss what steps to take at that point to further my career.