Did you change pg_hba.conf?

sudo nano /etc/postgresql/9.3/main/pg_hba.conf

and allow connection by all users to all databases from any IP if the user has right MD5-encrypted password by adding the following line:

host all all 0.0.0.0/0 md5

I have a step-by-step instruction in my blog (sorry for the shameless plug)

@kirilscherbach you've provided a much better instruction than the official page of Digital Ocean. The problem is that you haven't enabled the right url slug etc " How to configure your Postgresql database on Digital ocean" or "How to bla bla bla" and it would have saved me time searching for how to do exactly what you wrote.

You've done a fine job writing that article. Hope you change your url slug. ;)

Links above appear to be dead. This worked for me:
Connecting to PostgreSQL on Linux

I do the samething and works when i change password

sudo -u postgres psql

ALTER USER sammy PASSWORD 'newpassword';

If you are using firewall don't forget to allow port 5432
for ufw use :

ufw allow 5432/tcp