This can be done. Following are the steps to setup the GUI
sudo useradd -m awsgui
sudo passwd awsgui
sudo usermod -aG admin awsgui
sudo vim /etc/ssh/sshd_config # edit line "PasswordAuthentication" to yes
sudo /etc/init.d/ssh restart
In security group open port 5901. Then ssh to the server instance. Run following commands to install ui and vnc server:
sudo apt-get update
sudo apt-get install ubuntu-desktop
sudo apt-get install vnc4server
Then run following commands and enter the login password for vnc connection:
su - awsgui
vncserver
vncserver -kill :1
vim /home/awsgui/.vnc/xstartup
Then hit the Insert key, scroll around the text file with the keyboard arrows, and delete the pound (#) sign from the beginning of the two lines under the line that says "Uncomment the following two lines for normal desktop." And on the second line add "sh" so the line reads
exec sh /etc/X11/xinit/xinitrc.
When you're done, hit Ctrl + C on the keyboard, type :wq and hit Enter.
Then start vnc server again.
vncserver
You can download xtightvncviewer to view desktop(for Ubutnu) from here https://help.ubuntu.com/community/VNC/Clients
In the vnc client, give public DNS plus ":1" (e.g. www.example.com:1). Enter the vnc login password. Make sure to use a normal connection. Don't use the key files.
Additional guide available here: http://www.serverwatch.com/server-tutorials/setting-up-vnc-on-ubuntu-in-the-amazon-ec2-Page-3.html
Mac VNC client can be downloaded from here: https://www.realvnc.com/en/connect/download/viewer/
Port opening on console
sudo iptables -A INPUT -p tcp --dport 5901 -j ACCEPT
If the grey window issue comes. Mostly because of ".vnc/xstartup" file on different user. So run the vnc server also on same user instead of "awsgui" user.
vncserver