Can I use the same kind of tools to sharpen my knives as I do my chisels? I'm not sure if this would be better suited for here or the woodworking stack, so please migrate if needed. 

In order to sharpen my woodworking tools (chisels and hand planes), I currently use a DMT diamond plate (the fine and extra-fine, what they call 900 and 1200 mesh). I plan at some point to get a ceramic water stone (e.g., Shapton), probably a 5000 grit to start.

My process involves flattening (or polishing) the back of the chisel / plane iron and then flattening / sharpening the bevel on the front. The process is seen here in a video by Rob Cosman, although he uses a much finer stone.

Is this procedure (that is, diamond plate potentially followed by water stone) appropriate for kitchen knives? Obviously kitchen knives don't have a flat back like chisels do, so both sides of the knife would get a bevel.

My knives are mostly Wusthof classic, and use the chef's knife and santoku most often.
Yes. ~1000 then 5000 is a common progression to get kitchen knives to a good working sharpness. Diamond plates can leave undesirable deep scratches, so I would not recommend someone inexperienced in their use (note that the OP is by his own statement NOT inexperienced) to take them to an expensive knife. 5000 might be slightly too fine for a Wusthoff (depending on the vintage and exact steel/temper used) - if the edge turns out unstable, try to omit the 5000.

By the way, some traditional japanese cooking knives (eg sashimi knives) ARE sharpened in a way very similar to you chisels.