javascript onload calling a function of a plugin

I did not test this, but something like this should work. Edit the function all_js in your plugin file and add these lines:

$.post("/wp-admin/admin-ajax.php", {
  idPost: $(".post-read").data("id"),
  action: "read"
}, function(response) {
  // Handle errors etc.
});

You also need to ensure that the button data changes when the AJAX request is made.

Here is an improved script (still far from perfect, but an improvement):

jQuery(function($)
{
    var button = $(".post-read"),
        postId = button.data("id");

    // Updates button attributes to match its data-action attribute
    function updateButton() {
        if(button.data("action") == "unread")
        {
            button.attr("src", "http://corentinbuet.fr/wordpress/wp-content/plugins/Read-Unread/LU.png");
            button.attr("title", "Marquer comme Non lu");
        }
        else
        {
            button.attr("src", "http://corentinbuet.fr/wordpress/wp-content/plugins/Read-Unread/NON_LU.png");
            button.attr("title", "Marquer comme Lu");
        }
    }

    // Mark post as read
    $.post("/wp-admin/admin-ajax.php", {
        idPost: postId,
        action: "read"
    }, function(response) {
        // Handle errors etc.
        button.data('action', 'unread');
        updateButton();
    });

    $(".post-read").click(function()
    {
        // Fix: Declare variable
        var action = button.data("action");

        var data = {
            "action": action,
            "idPost": postId
        };

        $.post("/wp-admin/admin-ajax.php", data, function(response)
        {
            // Check for errors etc
            // Update button
            button.data("action", (action == "read") ? "unread" : "read");
            updateButton();
        });
    });
});