The best advice I can give someone who wants to work at the UN is to learn a second language (preferably one of theirs), try to get an advanced degree, and do some type of international work in the field that you want to get into even if you have to volunteer and pay your way. —ie. if you want to do humanitarian work in africa, volunteer w/ international organisations NOW that do those types of things (Red Cross, Habitat for Humanity, etc.). The UN has volunteer opportunities and they also have an intern program.
If you are American, remember that most people from other countries know two to three languages and have travelled. A degree isn’t going to cut it. Do something that says you have enough initiative and courage and makes you stand out amongst the best. Keep yourself current with international politics.