Running X Applications Remotely with Xming and putty
Xming is described as "the leading X Window Server for Microsoft Windows" and is available for free download. You will probably want at least the server and the server fonts installers - the default installation options are generally sensible too.
Xming Configuration
Xming comes with its own version of putty built in - using this may keep things simpler, but you may choose to use a standalone version of putty (or another SSH client altogether).
By default, Xming will be installed in "Program Files\Xming". If you run the XLaunch executable you will get a number of configuration options - including how you wish to display your X screens and whether you wish to launch a program on a remote computer.
If you select the option to launch a program, e.g. xterm, and select "Run Remote" and use the built in Putty (plink.exe) option, you can then select one of the School's compute servers (e.g. tinky-winky.cs.bham.ac.uk) to connect to, enter your user name & password you should then be able to launch the program on the server but view it locally on your computer. You can give additional options on the next screen if required.
If you choose to use a standalone version of putty or a different SSH client then you will need to enable X11 forwarding and tunnel X11 connections. There are some generic putty instrucions at http://gears.aset.psu.edu/hpc/guides/xming/