It is not possible to travel by bus between these two places entirely along the Mekong, in one day. Even though it looks easy and fast enough, expect about 8-9 hours of travel time, including a 3-4 hour departure from the riverside highway south and then back north to Chiang Khan.
Apparently, in the early morning there is a songthaew connection in Pak Chom that will deliver you to Chiang Khan, but you can't make it in time if you leave from Nong Khai, even on a 7:00 am bus. You might make it if you stay in Sangkhom (at least you used to be able to ... I did it several years ago).
Instead, without changing buses, you leave Route 211 at Pak Chom and travel south through the hills on Route 2108. You will have to hop off the bus in Phong (it may be called Ban Haad ... that's what the lady on the bus told me, but my map book says "Phong"), then cross the road (Route 201) and wait for a songthaew to Chiang Khan in a sala beside the police station. You won't have to wait more than 30 minutes, or so.
This route adds about 3 hours or more of time onto the trip.
A much quicker and more enjoyable way to do this is to make a loop out of part of the route. Hire a motorbike in Nong Khai and follow the river road west out of town. This road will eventually merge with Route 211, and will take you through Tha Bo (about 45 minutes), Si Chiangmai (about 90 minutes), Sangkhom (about 2 hours), Pak Chom, and all the way to Chiang Khan.
On the way back, travel south towards Loie from Chiang Khan on Route 201 and turn left onto Route 2108 that will take you to Pak Chom, then continue back to Nong Khai.
If you want to risk it, it's possible to hitch between Pak Chom and Chiang Khan, though there is very little traffic along the road. However, having said that, it's unlikely you'd be passed by very often. Folks in this part of the country are exceedingly friendly and helpful.
This is a spectacular area. The scenery along the Mekong is gorgeous, and even the extra time on the bus can be rewarding considering the beautiful landscape that surrounds you.