A curious question
There are no interesting (from a touristic point of view) places along the route and the roads are mostly in terrible shape. There is no company that is driving that route in one trip and nor will there be any willing to do that in the near future.
Only way is to change buses several times and the whole trip would take you several days (and probably several years of your life). You can be sure to find real local (mini)buses (not the local type that goes back and forth between PP and SR or HCMC which are riding palaces compared to that). Unless you are an extremely seasoned traveler you should not attempt this.
Either take a northern route via southern Laos (several days) or via Phnom Penh. Fly out from there or go overland to HCMC and then north (also several days). Northern route is shorter and less comfortable and the southern route is longer but more comfortable.
I personally don't see the point of coming all the way to South east asia and then spend half of your time on buses covering such distances.
Enjoy anyway. Let us know if you've done this. Would be nice to read a real travel story
I'm with eastwest. Just a note on the northern option - you'd have to get to Phnom Penh then you could get a direct bus from there to Pakse in Laos (check re visa but I assume you can get it on arrival - but bear in mind this is a significant added cost to your overall journey) then there's a direct bus from there to Hue. BUT after 7 months travelling in the region, that bus was the worst travel experience we had had: 16 hours door-to-door, crammed into a battered minibus, feet on sacks of rice, no head rest, people smoking, music blaring all night, dangerous driving....you get the picture! Of course, it was an experience and soon forgotten about but depends on your tolerance levels as to whether you're up for it! Anyway, it's going to take about 3 days all in. The route via HCMC would be more comfortable - although perhaps longer and more expensive - as you could take the the train from HCMC to Hue. Takes about 20+ hours. You might have to wait a few days to be able to get a ticket though. Good luck!