Just to add a little flavor to the basics that Marty has outlined below, attending the field school is the best way for TIGHAR veterans, and in particular Ric, to evaluate how you handle being in the field under what will be stressful and uncomfortable conditions. Nothing brings the worst out of a person than difficult circumstances, and Nikumaroro always presents difficult circumstances.
One thing that some people forget is that any expedition has only one Chief, and the rest of us are worker bees. If you aren't willing to buckle down and be a worker bee, stifling your desire to be a Chief, you won't do well in the world of expeditions.
Having a specialized skill that is needed by the expedition is another way to gain a spot, and those tend to be medical degrees or technical skills for some particular part of the work that is needed. One example is when we wanted to try out ground penetrating radar. We needed a trained GPR technician, but with no GPR specialist found within the TIGHAR membership, we had to look outside members and Field School graduates to find one.
Figuring out some angle of the Earhart project that you can particularly research, be it sextant numbers, archival research, knives, shoes, buttons, compacts, glassware or some similar productive research is also a good way to get noticed and may land you on the EPAC, but won't get you onto an expedition until the membership and Field School requirements are completed and you fit the profile for a good expedition team member.
One exception to the above is the Sponsor Team Member category. There are usually a few berths available for STM as a result of the reality of fund raising for expeditions. Ric vets STMs pretty carefully and thankfully most if not all (at least for the expeditions I've been on) have proven to be hard working contributors to the trips.
Finally, let me just say that dogged persistence and determination help. It took me some 10 years from initial membership to get onto the Niku expedition roster, so you can't take a short term approach if going to Niku is one of your goals.
Andrew