How can I connect to a web database from my PC?

You can connect with PHP to both local and external databases without problems, just by indicating the IP address of the database (and in some cases the port). The real problem you are going to face is telling the database to accept connections from remote users (shared server databases usually restrict access to only local users).

As you mention, you can use a program like WAMP to have a local server (Apache+PHP) and from there run the PHP code which will connect to the remote database. If you choose this option, it is likely that you will have to send a ticket to the server support (if the server offers support for these cases) asking them to enable the connection of remote users to the database (they will probably ask you for your IP for only unblock the connection from your IP and not for remote users in general).

The other option (and more recommended) is to work directly on the remote server. That is, upload the PHP files to that server and do the programming there. This gives the advantage that you will be connecting to the database as a local user (because the PHP script you upload will be on the same remote server as the database).

You can see a small article that talks about how to connect to a database to retrieve information:

http://blog.aulaformativa.com/connect-php-with-mysql/

Remote connections depend on both parties being generous when working, let me explain. On the server side where the DB you need to access is located, it must allow remote connections, so it normally has to be configured to allow the IP of the computer from which you want to access. Well, now on your side, or the remote client, what you need is access through some handler. I show you an image with Query Browser:

Having the data you claim to have, you can access without problems. All the best.

I share a class to make the connection to the database, this connection is made through php :

<?php

  # Clase conexion base de datos Mysqli
  # Bases de datos Mysql en php
  # PHP 5.0



class MySQL
{
  $conexion;
  $host     = "ipDelHost";
  $user     = "Usuario";
  $password = "Contrasena"
  $dbname = "NombreBaseDeDatos";

  function MySQL()
  {

  if(!isset($this->conexion))
    {
        $this->conexion = mysqli_connect($this->host,$this->user,$this->password);
      if (!$this->conexion) {
        die('error en la conexion: ' . $mysqli->connect_errno);
        exit();
      }

         mysqli_select_db($this->conexion,$this->dbname) or die("error en la conexion de base de datos: ".mysql_error());
    }
  }
/* retornará un objeto (mysqli_result) con el resultado de la consulta */
 function consultar($query)
 {
    mysqli_query("SET NAMES 'utf8'"); 
    $resultado = mysqli_query($query,$this->conexion);
    if(!$resultado)
    {
            echo 'MySQL Error: ' . mysql_error();
        exit();
    }
  return $resultado; 
  }
/* retornará "true" si el registro es exitoso */
  function registro($query)
  {
      $resultado = mysqli_query($query,$this->conexion);
    if(!$resultado)
    {
            echo 'MySQL Error: ' . mysqli_query();
        exit;
    }
    return $resultado; 
  }

 /* retornará un array con los datos de la consulta*/
 function fetch_array($query)
 { 
    return mysqli_fetch_array($query);
 }

/* Retorna el número de filas del resultado. */
 function num_rows($query)
 { 
     return mysqli_num_rows($query);
 }

 /* Retorna un array de cadenas que se corresponde con la fila obtenida (como una matrix) $fila[0], $fila[1]*/
 function fetch_row($query)
 { 
     return mysql_fetch_row($query);
 }

 /* Devuelve un array asociativo de strings que representa a la fila obtenida del conjunto de resultados, donde cada clave 
    del array representa el nombre de una de las columnas de éste; o NULL si no hubieran más filas en dicho conjunto de resultados.

    Si dos o más columnas del resultado tienen el mismo nombre de campo, la última columna tomará precedencia. Para acceder a la/s otra/s 
    columna/s con el mismo nombre, es necesario acceder al resultado ya sea usando los índices numéricos mediante mysqli_fetch_row(), 
    ya sea añadiéndole alias a los campos.

    Ejemplo

    $consulta = "SELECT Name, CountryCode FROM City ORDER by ID DESC LIMIT 50,5";

    # obtener array asociativo 

    while ($row = mysqli_fetch_assoc($resultado)) {
        printf ("%s (%s)\n", $row["Name"], $row["CountryCode"]);
    }
  */
 function fetch_assoc($consulta)
 { 
     return mysql_fetch_assoc($consulta);
 } 

}

To use it, you create the class object and call the function you need by sending the query, there it tells you that each function returns, I leave you an example:

<?php
/* llamas la pagina donde esta la clase de conexión */
require_once 'MySQL.php';

/* creas el objeto de la clase */
$mysql = new MySQL();
/* realizas la conexión */
$mysql->MySQL();

$query = "SELECT Name, CountryCode FROM City ORDER by ID DESC LIMIT 50,5";
/* realizas la consulta */
$mysql->consultar($query)

I hope it helps you