How do I update core with Composer?
To update Drupal using Composer, you just need to move the "drupal/core": "~8.1" line from the replace section to the require section. At the end the content of the composer.json file is like the following one.

{
    "name": "drupal/drupal",
    "description": "Drupal is an open source content management platform powering millions of websites and applications.",
    "type": "project",
    "license": "GPL-2.0+",
    "require": {
        "composer/installers": "^1.0.21",
        "wikimedia/composer-merge-plugin": "~1.3",
        "drupal/core": "~8.1"
    },
    "minimum-stability": "dev",
    "prefer-stable": true,
    "config": {
        "preferred-install": "dist",
        "autoloader-suffix": "Drupal8"
    },
    "extra": {
        "_readme": [
            "By default Drupal loads the autoloader from ./vendor/autoload.php.",
            "To change the autoloader you can edit ./autoload.php."
        ],
        "merge-plugin": {
            "include": [
                "core/composer.json"
            ],
            "recurse": false,
            "replace": false,
            "merge-extra": false
        }
    },
    "autoload": {
        "psr-4": {
            "Drupal\\Core\\Composer\\": "core/lib/Drupal/Core/Composer"
        }
    },
    "scripts": {
        "pre-autoload-dump": "Drupal\\Core\\Composer\\Composer::preAutoloadDump",
        "post-autoload-dump": "Drupal\\Core\\Composer\\Composer::ensureHtaccess",
        "post-package-install": "Drupal\\Core\\Composer\\Composer::vendorTestCodeCleanup",
        "post-package-update": "Drupal\\Core\\Composer\\Composer::vendorTestCodeCleanup"
    },
}


I was used to run drush upc to update Drupal, but that overriden my composer.json file, and the vendor directory. As consequence of this, I should run composer update all the times to update the content of the vendor directory.

Now, using Composer to update Drupal, I can also use it to download/update the modules I am using.

The composer.json file I am using is the following one.

{
    "name": "drupal/drupal",
    "description": "Drupal is an open source content management platform powering millions of websites and applications.",
    "type": "project",
    "license": "GPL-2.0+",
    "require": {
        "composer/installers": "^1.0.21",
        "wikimedia/composer-merge-plugin": "~1.3",
        "drupal/core": "~8.1",
        "drupal/imagick": "8.1.*@dev",
        "drupal/mollom": "^8.1",
        "drupal/honeypot": "^8.1"
    },
    "minimum-stability": "dev",
    "prefer-stable": true,
    "config": {
        "preferred-install": "dist",
        "autoloader-suffix": "Drupal8"
    },
    "extra": {
        "_readme": [
            "By default Drupal loads the autoloader from ./vendor/autoload.php.",
            "To change the autoloader you can edit ./autoload.php."
        ],
        "merge-plugin": {
            "include": [
                "core/composer.json"
            ],
            "recurse": false,
            "replace": false,
            "merge-extra": false
        }
    },
    "autoload": {
        "psr-4": {
            "Drupal\\Core\\Composer\\": "core/lib/Drupal/Core/Composer"
        }
    },
    "scripts": {
        "pre-autoload-dump": "Drupal\\Core\\Composer\\Composer::preAutoloadDump",
        "post-autoload-dump": "Drupal\\Core\\Composer\\Composer::ensureHtaccess",
        "post-package-install": "Drupal\\Core\\Composer\\Composer::vendorTestCodeCleanup",
        "post-package-update": "Drupal\\Core\\Composer\\Composer::vendorTestCodeCleanup"
    },
    "repositories": {
        "drupal": {
            "type": "composer",
            "url": "https://packagist.drupal-composer.org"
        }
    }
}


Apart from moving that line as I said, I also executed the following commands.

composer config repositories.drupal composer https://packagist.drupal-composer.org
composer require drupal/honeypot
composer require drupal/mollom
composer require drupal/imagick 8.1.*@dev


The first allows me to require Drupal modules and update them.

The last time I tried, the Drupal official repository was still in alpha stage, and it gave me problems with modules that declared themselves as Composer packages, but not setting the package type (i.e. drupal-module). I hope they fixed the problem, now. For the Drupal Packager repository, the commands to use are the following.

composer config repositories.drupal composer https://packages.drupal.org/8
composer require drupal/honeypot
composer require drupal/mollom
composer require drupal/imagick 1.*@dev


If you were already using https://packagist.drupal-composer.org, and you want to replace with the official Packagist site, you need to replace the first command with composer config repositories.0 composer https://packages.drupal.org/8, which works if you don't have other repositories, basing on Using packages.drupal.org.

Notice also that packagist.drupal-composer.org is scheduled to be deprecated in January 2017 (or when the official Package Repository from Drupal.org is ready).