Showing a partial view in a modal popup

With your current code, when user clicks on the submit button, it will do a normal form submit as your submit button is inside a form tag. For your use case, you should be hijacking that normal form submit event using javascript and make an ajax call to your action method where it will use the search_type and search_string parameters to get the filtered data and return a partial view result. This partial view result is the HTML markup you want to display inside the modal dialog. Once your ajax call receives the response from server, update the modal dialog's body content with this response and fire the modal dialog.

@using (Html.BeginForm("Index", "Accounts", FormMethod.Post, new { id = "searchForm" }))
{
    <div>
        <input type="text" name="Search_String" />
        <input type="submit" id="submit" value="Search" />
    </div>
}    
<div class="modal fade" id="myModal">
    <div class="modal-dialog">
        <div class="modal-content">
            <div class="modal-header">
                <button type="button" class="close" 
                                      data-dismiss="modal" aria-label="Close">
                    <span aria-hidden="true">&times;</span>
                </button>
                <h4 class="modal-title">Modal title</h4>
            </div>
            <div class="modal-body">
            </div>
            <div class="modal-footer">
                <button type="button" class="btn" data-dismiss="modal">Close</button>
                <button type="button" class="btn btn-primary">Save changes</button>
            </div>
        </div><!-- /.modal-content -->
    </div><!-- /.modal-dialog -->
</div><!-- /.modal -->

Now have some javascript code, which listen to the submit event on your search form and make stop the normal behavior (normal form submit) and instead do an ajax form submit.

$(document).ready(function () {

    $('#searchForm').submit(function (e) {
        e.preventDefault();
        var $form = $(this);

        $.post($form.attr("action"), $form.serialize()).done(function (res) {
            $mymodal = $("#myModal");
            //update the modal's body with the response received
            $mymodal.find("div.modal-body").html(res);
            // Show the modal
            $mymodal.modal("show");
        });
    });

});

Now you have to make sure that your Index action method returns a partial view (so that it will not execute any layout code, but just that view code).

[HttpPost]
public ActionResult Index(string Search_Type, string Search_String)
{
    // Your existing filtering code goes here.
    return PartialView(accounts.ToList());
}