The Marshalls are tiny and remote, and consequently flight connections (specifically, to Majuro, the only airport of any size) are extremely limited. Getting there from Bangkok will involve at least two hops.
The only two airports with longer-distance connections to Majuro are Guam (GUM) and Honolulu (HNL). Your least bad option will thus likely involve ...
According to the Permanent Mission of the Republic of the Marshall Islands to the United Nations,
All visitors must have a passport valid for at least six months, must have sufficient funds for stay, and have an onward/return air or sea ticket.
United flies 3 times a week non-stop from Honolulu<->Majuro. That's about $1500,- round trip.
BKK <-> HNL round trip will run you about $1000 (ANA in January) and as little as $750 if you are willing to deal with crappy routing on Korean.
So you should be able to do this for $2500 or less, but you need to buy two separate tickets and make sure you ...
TL;DR About $2000, and Brisbane!
The two sites best for working out your options are
Flightconnections.com which shows you a map of nonstop routes
FlightMapper.net which gives you a list of direct flights
In this particular case, looking at the Majuro connection list from FlightMapper
we see that there is a direct flight to Nauru, and checking the ...
There is a island hopper flying between Guam and Honolulu 3 times per week (in each direction). This flight stops in the Marshall Islands and a few other places (Chuuk, Pohnpei etc). I went to Pohnpei last year on this flight and it is a rather straightforward flight, the airplane is actually a Boeing 737 (would have expected something smaller). Flight ...