Grey pixelated screen when logging into server via VNC

Your image link is broken but based on what you’ve described I think I know where you’re running into trouble. The current Gnome desktop is known to have issues with some remote desktop software and for it to work best, hardware accelerated graphics (which are not available on a droplet) is recommended.

The grey screen you are seeing is likely the default X.org view either because you did not include a line to launch a gnome-session in your xstartup file or because gnome was unable to start.

An alternative to VNC is to use X2go which generally provides better performance and tunnels over ssh by default so your connection is encrypted. From a clean 16.04 droplet this script will install and start x2go with xfce:

#!/bin/sh
#
# This script will set up an xfce desktop environment which can be
# accessed remotely via the x2goclient http://wiki.x2go.org/doku.php/doc:installation:x2goclient
export DEBIAN_FRONTEND=noninteractive;
apt-get update;
apt-get -y install software-properties-common xubuntu-desktop;
add-apt-repository -y ppa:x2go/stable;
apt-get update;
apt-get -y install x2goserver x2goserver-xsession;