The expression km de route does exist in French and Google finds many occurences but Chaiyaphum est à 337 km de route au nord-est de Bangkok is not something I would expect to hear from a native French speaker. What would be more common is: Il y a 337 km de route pour aller de Bangkok à Chaiyaphum but it is difficult to place northeast in this sentence.

Here is what I can suggest:

Chaiyaphum se trouve au nord-est de Bangkok, à 337 km par la route.

This is close to what I found here:

La distance entre Bangkok et Chaiyaphum est de 282 km. La distance par la route est de 337.5 km.

"Kilomètres de route" is completely correct. As a native French speaker, I use it on a daily basis.

We tend to consider it the normal way to express a distance though, and so usually omit it. The opposite being "kilomètres à vol d’oiseau", is also frequently used.

Suggestion from Gilles Thésée:

“kilomètres par la route”, as opposed to “kilomètres à vol d’oiseau” (“as the crow flies”)

https://www.google.com/search?q=%22kilom%C3%A8tres+par+la+route%22 does return a few results (but not that many).