Airfares to and from Bangkok are generally fairly cheap in the first place. However, there are a few tips to make sure your airfare is as low as possible.

Travel Off-Season. High season to Thailand is usually October to April when airfares are high. And they get super high during Christmas and Songkran (Thai New Year). The summer months are low-season so airfares to and from Thailand are cheaper. Especially May/June and September are months where airlines discount their normal airfares significantly.
 
Travel light. Some airlines are charging for extras and these extras may include luggage that is places in the aircraft’s cargo hold. So if you want to avoid these extra fees, just pay the airfare and just bring carry-on luggage.

Travel low-cost. Some of the lowest airfares in Thailand are published by low-cost airlines. That’s all well and good but beware that these airlines are knows to charging you lots of extra fees on top of your airfare. So just shop around and check airfares from many different airlines to make sure you get the best deal.

Plan early. All airlines offer cheap airfares, however, there is just a certain number of seats of each flights being made available for the lowest airfare. Once this capacity has been sold out, the airfare moves up. So the earlier you book, the cheaper normally the airfare. This rule is particularly important for Thailand because many visitors to Thailand come from places quite far away, hence they tend to plan and book their Thailand holidays early.
 
Travel with a stop. Airfares on nonstop flights are usually more expensive than airfares on connecting itineraries. In the case of Thailand, try connecting in the Middle East if you fly to Europe or in Korean/Japan if you want to travel North America.