Taxi.
Get in the official taxi stand queue just outside the airport doors and give the person at the counter the address you're going to. They will tell you how much it will cost (or possiblymore likely the ride will be metered - it won't matter much either way, as Bangkok taxis are cheap). They will give you a docket, and point you towards the taxi driver who will take you.
Politely ignore any hawkers who try and convince you to go with them in their private car or "taxi". They'll want a higher price. Just go to the official taxi stand queue.