With its lack of a Skytrain (BTS) or underground (MRT), the easiest way to reach Chinatown - if you want to avoid the traffic - is by boat (Chao Phraya River Express).
Get off at the Ratchawong Pier, and walk up from Ratchawong Road to Sampeng Lane and Yaowarat Road. Ordinary buses 1, 7, 8, 37, 49 and 75 will take also take you here, but it's much better to travel by boat. The Ratchawong Pier is just a few hundred meters from Yaowarat Road and Sampeng Lane.
You'll know that you've arrived when you reach the Odeon Circle, where a huge ceremonial Chinese gate unmistakably marks the entrance. Chinatown itself is an easy to explore on foot and, in fact, there really isn't any other way.
An alternative to get to Chinatown - and to avoid at least some of the traffic - would be to take the underground (MRT) to Hualamphong station, and either a taxi or tuk-tuk from there. For more information on how to get around in Bangkok, have a look at our comprehensive Getting Around in Bangkok Guide.