Extending range is not a big problem with the TC or any apple wireless router.. there is a specific function to do it.. although mixing the new AC model into the older N ones has been giving people some issues.
So you have two choices.. wireless extend.. or use ethernet or homeplug (EOP) adapters to connect them as a roaming network.. the later is much the better with ethernet at least.. homeplug has the same issues as wireless.
You cannot put the TC into the dead wireless spot. You need it half way so it gets good signal and can extend it.
Apple have loads of info but the stuff is out of date cf the new 6.3 utility.
Look up extend wireless .. it is about 10% of all posts over in airport area..
Here is one for setting up roaming on 6.3 https://discussions.apple.com/thread/5134082?tstart=0
Apple doco on extend.
http://support.apple.com/kb/HT4145
The rest of the setup is same.. both will be available on both computers and you can backup to either.
It is hard to start with nothing.. read the info available and when you get to the setup yell if you have problems.
But you need to decide how you want to do it. And learn how to use the 6.3 utility.
The actual setup of extend.. is simplicity itself. If everything works right.
You just factory reset the old TC so the computer can pick it up.. whilst it is connected to the new TC by wireless or ethernet. You then get a setup wizard via the utility that will step you through joining the wireless and putting in the password.. voila. takes one min. The setup wizard will configure the old TC correctly.. and that is the best way to do it.