The most common scenario when running an X client is that the X server is on the workstation which has the user console. That means either the X server is on the same host as the X client, or that the user has logged into the X client host from a workstation that is running the X server. I will assume the latter since there are two OS's involved.
The easiest (also the best & most secure) way to accomplish your objective is to log into the X client linux host with an SSH client that can do X tunneling. When doing so, it will automatically set up the correct $DISPLAY variable on the X client host, and will silently route all X traffic to the X server on your local host. For Windows SSH clients, putty
seems to be the weapon of choice for most people, as I understand it. SSH under Cygwin will also work, and there are a number of others.
If you were doing this from another Linux shell, the ssh command would look like
ssh -X firstname.lastname@example.org
The method of manually setting $DISPLAY on the X client host may not work at all unless the X server is configured to accept remote TCP connections. Using the X tunnel method, the X traffic is seen by the X server as local, since the local ssh client is what makes the X connection.