I'd either attack this project with dominoes -- dominoes alone -- (for speed, simplicity, and efficiency), or do it in the traditional way (rabbets and dados). There's little to gain from using the dominoes for the top/bottom and sides only, and then some other joinery for the verticals.
Info. on what is to be put in the unit is not known. For average objects, the dominoes reinforced with the verticals nailed or dominoed to the back should be good enough.
If the idea is to fill every shelf with LPs or heavy items, I may be worried about sacking whether you're using dominoes or not. I'd also not use just one single French cleat, but two, top and close to the bottom.