One way you could do it without removing your hard drive is by creating one of the Ultimate Boot CDs for Windows
found at this site:http://www.ubcd4win.com
It is basically a Windows OS that runs on a CD. If the hard drive is in good enough condition to recover, you could use that. But it is probably more complicated to walk you through that than it is to take the hard drive out.
But if you want to...
Only other way is by putting it in a working computer, or puting a working hard drive in yours as primary. Or...actually, you can buy a USB external hard drive adaptor
. Basically turn your internal hard drive into an external one. That's another option. But I think it would be better to stay away from spending money. Things aren't looking too good for your hard drive right now, according to that message you got from Chkdsk.
I have the broken computer hooked up to a working computer using our home network, can that be used?
Only if you create an Ultimate Boot CD for Windows
or if you put a hard drive from someone's computer that works. Other than that, no.
there anyway I can transfer data between computers using a CAT5 cable or something like that?
Need a working operating system to do that. So once again, only with an Ultimate Boot CD for Windows
or a working hard drive with an operating system. Other than that, no.