I am a client on a weak wifi network that I can only pick up from one
room in my apartment with my desktop PC. Is it possible to share that
connection with my own wireless router to allow the other computers in
my home to access it? (In effect, extending the signal). I am not
interested in over the counter 'repeaters' because I would like to
create a secure private network on my end and use the unsecure network
as the internet connection (instead of just making the unsecure
network larger). Would it be as simple as plugging the router into the
LAN port of the pc and changing some settings to link them? Be as
detailed as you can on how to accomplish this, including any extra
hardware or other ideas you may have. THANKS!

You could take a wifi/ethernet bridge similar to this:
http://froogle.google.com/froogle?q=D-Link+DWL-G810&btnG=Search+Froogle

and connect its ethernet port to your wireless router's WAN/Internet
port. These bridge boxes will just snatch up the wireless and put it
on the wire, and vice versa. Then, set up your router to get DHCP from
the WAN side, and you can secure your LAN/wifi side as desired. A bit
ugly but it should work!

-jason

My recommendation would be to set your router to act as a client. 
Linksys routers offer this functionality, for instance.

That way, you would bind to the wireless network via your wireless
antennae and the router would provide service to that network
(functioning as an Internet connection) over the wireless connection,
rather than the "Internet" port.

Also, if the signal is weak, 7dBi antennae might be recommended, as
they would increase the signal strength by ~166% and are even
available at WalMart.