The really Cheapest flights Hawaii to Bangkok or Bangkok to Hawaii are usually on either China Air or Eva Air both airlines are based out of Taiwan China. If you are flying on a one way ticket to Bangkok you need to have either a transit visa or another flight out of Thailand otherwise there is a good chance that they will not let you board the flights as Thailand Might refuse you entry if you don't have a outbound ticket I never heard of it happening, but the airlines check for it as they are ones that get stuck flying you back to Honolulu for free. To get the best prices you need to book the flights at least a month in advance.
Your best bet is to try the large travel sites like Expedia or Travelocity once you have a idea on the prices then try Side Step which is a new service that uses over 100 travel sites to get the best rates.