You would open your ECAD of choice, draw the schematics, import it into the PCB, maybe define a boardstack of a flexible PCB if you feel like it. Then you look up the design rules you should use for flexible PCBs from your PCB house of choice. You layout your LED strip and order the flexible PCBs, then you just have to solder the components to your flex PCB and you are done.
Seriously though, you go about it just like a normal PCB but you order a flex PCB instead.