The "Death Railway" was constructed between 1941 and 1943 during World War Two with the intention of connecting Thailand and Myanmar (Burma) by rail so that Japanese forces, who had recently invaded Thailand, could move troops, equipment, and weapons into Myanmar and push the allied forces back into India and beyond.

I had learned about the history of the Death Railway in school, but I didn't fully understand it until I watched The Railway Man, a movie starring Colin Firth and Nicole Kidman, on a flight last year. The history and train nerd in me knew I had to go visit it when I first landed in Thailand a week ago.

The Thailand-Burma Railway is used for travel.
The Death Railway itself begins in Nong Pladuk, some 50 kilometers to the west of Bangkok, where it connects to the express highway running from Bangkok to Singapore. It slowly meanders from Nong Pladuk to Kanchanaburi, the location of the famed Bridge Over the River Kwai, through dust bowl suburbia. The panorama gradually improves once the train over the fabled bridge and river, with jungle, mountains, and the twisting river to the left. As it prepares to creak over the pack of cards wooden bridge, also known as the Wampo Viaduct, the train slows to a stop.

How to Get There, How to Get There Fast, and Where to Stay
The first choice would be to spend a day traveling from Bangkok to tour the Thailand Burma Railway. You may see the complete route by taking the first train from Bangkok to Nam Tok, visiting Hellfire Pass there (you'll need to take a Song-Thaew or taxi from Nam Tok), and then returning to Bangkok the same day by taking the afternoon train. It takes roughly four and a half hours to go from Bangkok to Nam Tok.

This alternative involves taking a long day's worth of trains (9 hours if you travel both ways) and leaves you with no free time to visit Kanchanaburi. If you only have one day, I'd recommend going with this option. If you have more time, I would suggest switching to...

The second option is to travel to Kanchanaburi by train from Bangkok Noi and stay there for one or two nights. Take the early-morning train from Bangkok Noi to Kanchanaburi on the first day. Here you can find accommodations in Kanchanaburi. I slept at the reasonably priced VN Guesthouse on the river, which offers stunning views from the restaurant and some rooms that are floating on the water, which is incredibly soothing. Explore Kanchanaburi in the afternoon of day one, go take pictures of the Bridge over the River Kwai, stroll through the war cemetery, and head to the Death Railway Museum in the city.

Take the morning train from Kanchanaburi to Nam Tok on day two, where you can tour Hellfire Pass (as per the one-day itinerary) before taking the train back to Kanchanaburi or, if you're not staying an extra night, to Bangkok. This, in my opinion, was the best method for traveling on the Thailand–Burma Railway.
To aid in additional preparation for this journey, the ever-helpful Man in Seat 61 has a wealth of information and train schedules.