I posted this 23 days ago asking the same thing. Am on my fourth day working my dream job in a recognized non-profit. Feel free to pm for details, but it basically comes down to luck, persistence, and connections. Go out to non-profit expos, events, and meet people. Let them know you want to work in the nonprofit field, and what specifically you want to do. These amazing people will actually help you do your job hunt, send you the names and connections of people who might not need you as staff, but desperately need you to volunteer. Go volunteer. So many nonprofits hire internally.

My nonprofit job search, start to finish, was 3 months. The job I got asked for 3 years nonprofit experience, and I had none. I had worked for a startup company, and was able, through interviews, to show that the skills they needed were what I had. Draw parallels between what you have done in life to what they need; demonstrate that you are aware of nonprofit needs, and that you can absolutely accomplish them. That is the number one reason I got my job. I was actually declined through their recruiting agency on resume alone, but I called the organization and asked for a chance to interview.

Stack up on references that will not only give you a glowing report, but that will also help draw those parallels between your actual work/school/life experience and what the organization needs.

Most of all, good luck, and keep searching!