Granted they have similar general form factor.. and if need to I think you can swap mobo. but the problem I'm seeing really is the mounting holes that's about it.
5c rear shell + 5s mobo + touchID home button + screen= 5c with touchID?
Yes, OP, I believe, in theory, it is entirely possible with what you are suggesting, but I recommend also switching out the battery and camera, and flash. There are just a few things though...
The casing to both the 5S and 5C have similar screw posts to mount the mobos on.
Image from iFixit.com.
The hole for the TouchID on the 5S is a little bigger than the hole on the 5C for the Home button, or else you would have been able to use the same screen (not the case). Just make sure the 5S display frame also comes with the display assembly.
The battery for the 5s is rated at 3.8V - 5.92Wh - 1560mAh, whereas the one in the 5c is rated at 3.8 V5.73 Wh1510 mAh; while the connectors are visually similar, I would advise swapping batteries to avoid problems.
The flex ribbon for the Power/Sleep/Volume/Vibrate assembly appears to have the same flex connector to the mobo, and at the same location, so you could potentially preserve the original flex intact. The flash is integrated to this flex, however, so if it does not work properly, you'll have to try "rigging" the 5s flex to the assembly.
iPhone 5C Power flex
Image from ProTechBuy.com
iPhone 5S Power flex
Image from PhonePartsUSA.com
The rear camera appears to have the same connector, as well. I would still recommend swapping it out.
Good luck if you attempt to do this! It's going to be one heck of an interesting franken-project.