Is it cheaper to book a return ticket or two one-ways?

Almost always cheaper to book a round-trip ticket from North America. One-way international flights are often priced at a premium by major carriers.

Should I use a third-party booking site like Expedia or Kayak?

Use them to find the itinerary, but book directly through the airline's website. If there is a flight cancellation or change, dealing directly with the airline is significantly easier than dealing with a third-party customer service line.

Finding Cheap Flights to Bangkok from North America

Use Google Flights to track prices 3-6 months in advance and look for departures from major hubs like LAX, SFO, or YVR. Focus on airlines like EVA Air, Cathay Pacific, or China Airlines, and be prepared to book a separate budget flight from a regional hub if it saves you more than $200.

Set a price alert on Google Flights. Enter your departure city and Bangkok (BKK) as the destination. Toggle the 'Track prices' switch. Set this at least 4 months before your intended travel date to get alerts when the fare drops below your threshold.
Search 'Anywhere' to find the cheapest gateway. If you don't live in a major hub, search for flights to Bangkok from your home airport and compare them to flights departing from LAX, SFO, SEA, and YVR. Often, booking a cheap domestic 'positioning flight' to one of these west-coast hubs can save you $300+ on the long-haul leg.
Check the 24-hour rule. If you book directly through an airline's website while in the US, you are legally entitled to a full refund within 24 hours of booking, provided the flight is at least 7 days away. Use this window to finalize plans without fear of immediate price hikes.
Compare dates in the calendar view. Once you have a baseline price, click the date field in Google Flights to open the calendar grid. Look for Tuesday and Wednesday departures; these are consistently $50-$150 cheaper than weekend flights for trans-Pacific routes.

Is it cheaper to book a return ticket or two one-ways?: Almost always cheaper to book a round-trip ticket from North America. One-way international flights are often priced at a premium by major carriers.
Should I use a third-party booking site like Expedia or Kayak?: Use them to find the itinerary, but book directly through the airline's website. If there is a flight cancellation or change, dealing directly with the airline is significantly easier than dealing with a third-party customer service line.