As mentioned above, the most obvious problem is that you don't have X
protocol forwarding set up (X11 is a client/server system where, quite
confusingly, a client is a program running sometimes on a remote
machine and it communicates with an X Server which you can think
of as a graphical system running on your local desktop).
You can set up Putty to forward your X connections but it is a bit
involved for a newbie. Instead, consider using MobaXterm which is
an out-of-the-box solution providing both X server and remote terminal
facility. With it, you can SSH into your Linux box and start X programs
("clients") which will show up on your desktop.