Where do I replace the sqli connect code

I am looking to see where I place the mysqli replacement code so that I can eliminate the connect errors I get...

function DBLogin()
{

    $this->connection = mysql_connect($this->db_host,$this->username,$this->pwd);

    if(!$this->connection)
    {   
        $this->HandleDBError("Database Login failed! Please make sure that the DB login credentials provided are correct");
        return false;
    }
    if(!mysql_select_db($this->database, $this->connection))
    {
        $this->HandleDBError('Failed to select database: '.$this->database.' Please make sure that the database name provided is correct');
        return false;
    }
    if(!mysql_query("SET NAMES 'UTF8'",$this->connection))
    {
        $this->HandleDBError('Error setting utf8 encoding');
        return false;
    }
    return true;
}

I have the following example, just not sure what I have to remove from the code above to get it to work. Any help?

<?php
$link = mysqli_connect("127.0.0.1", "my_user", "my_password", "my_db");

if (!$link) {
echo "Error: Unable to connect to MySQL." . PHP_EOL;
echo "Debugging errno: " . mysqli_connect_errno() . PHP_EOL;
echo "Debugging error: " . mysqli_connect_error() . PHP_EOL;
exit;
}

echo "Success: A proper connection to MySQL was made! The my_db database is    great." . PHP_EOL;
echo "Host information: " . mysqli_get_host_info($link) . PHP_EOL;

mysqli_close($link);
?>