A few tips:
1. Get a relevant Master's degree
2. Have relevant experience in government or a major NGO
3. Be willing to consider short term assignments and consultancies as a way of gaining familiarity with the UN and its methods (to see if you really want to work there) and also to get contacts and credibility, and to help hear about possible opening.
4. Use sites such as unjobs.org, unjoblist.org and devex as well as official UN sites to look for opportunities.
5. Some countries' permanent missions to the UN help their citizens find positions - try contacting them to see if they can help/advise you.
6. Be persistent. Expect to apply for a lot of jobs before something comes up. Repeatedly inquire about the status of your application - use contacts to find out opportunities are upcoming. Try to meet with or speak by phone to people who are in departments you might be interested to work in to get yourself known and to find out about possible opportunities.