First step is to check that you are running at least php 5.6 on your server (php -v)

Then perform the following steps through the terminal:

1 : php composer require magento/product-community-edition 2.1.0 --no-update
2 : php composer update

Once this is done, perform the step below
3: php bin/magento setup:upgrade

You may get the following error message :
..................requires magento/framework ~100.0.0 -> satisfiable by ..................

This may be due to :
"magento/data-migration-tool": "2.0.2",

You should temporarily remove this line at reguire in composer.json for this update, because at the time of creating this form item, the migration is
tool not yet compatible with magento 2.1

There also seems to be a bug that you get this message: Exception #0 (Exception): Warning: is_dir(): open_basedir restriction in effect. File(/etc/pki/tls/certs) is not within the allowed path(s): (/var.......

Change a line in composer.json :

"composer/composer": "@alpha"
 Naar:
 "composer/composer": "1.1.2 as 1.0.0-beta1",

After this, perform steps 1 , 2 and 3 again.

Also remember to set permissions properly after this update (chown -R)