It may depend on how lame (lack of ram, etc.) that old computer is.
With a decent Linux system, you can install cygwin on the Windows system, including the xwin program from cygwin, then use xwin on Windows to open a full Linux desktop as a window on your Windows desktop. I'm not sure what exactly you must install on the Linux side to accept X Windows connections, but if that is your plan, I'm sure many people here do know exactly what you should install.
For a less powerful Linux system (or a slower hardware link between the two systems, your much better of with putty). Again, the best start is to install cygwin on the Windows side and include putty in that cygwin install. Putty lets you open a Linux command window as a window on you Windows system. It is a lot faster and lower overhead than xwin.
Putty is good for compiling Linux programs, running Linux console apps and many other Linux tasks. It may be pretty lame for editing the source code (depending on what you like in an editor). You could install Samba on the Linux side and make your work directory in Linux accessible as a share in Windows, then use any Windows text editor you like.
Be aware that most Windows text editors work only with Dos carriage control (a CR before each NewLine). The compiler won't care if those CR's are in your source files, but you probably also want to edit Linux script files. Those will mess up with CR characters. You could use a Windows editor (such as emacs) that can be told to respect Linux carriage control, or you can use the dos2unix program to strip the CR's after editing script files.
You may also need to ask a follow up question of what to install/configure on the Linux side to accept ssh connections from putty.
Last edited by johnsfine; 06-18-2009 at 02:41 PM.