How to open UI of amazon EC2 instance.

This can be done. Following are the steps to setup the GUI

Create new user with password login

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

Setting up ui based ubuntu machine on AWS.

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: http://www.realvnc.com/download/get/1286/

With addition to this try this also :-

So I follow first answer, but my vnc viewer gives me grey screen when I connect to it. And I found this Ask Ubuntu link to solve that.

The only difference with previous answer is you need to install these extra packages:

apt-get install gnome-panel gnome-settings-daemon metacity nautilus gnome-terminal

And use this ~/.vnc/xstartup file:

#!/bin/sh

export XKL_XMODMAP_DISABLE=1
unset SESSION_MANAGER
unset DBUS_SESSION_BUS_ADDRESS

[ -x /etc/vnc/xstartup ] && exec /etc/vnc/xstartup
[ -r $HOME/.Xresources ] && xrdb $HOME/.Xresources
xsetroot -solid grey
vncconfig -iconic &

gnome-panel &
gnome-settings-daemon &
metacity &
nautilus & 

gnome-terminal &
References :- http://stackoverflow.com/questions/25657596/how-to-set-up-gui-on-amazon-ec2-ubuntu-server
http://www.serverwatch.com/server-tutorials/setting-up-vnc-on-ubuntu-in-the-amazon-ec2-Page-3.html