Displaying “alt” attribute text as captions?

A WordPress approach

If you have all your Images in the media gallery, you are able to access their post object while embedding/including the images via tinyMCE or plain PHP. WordPress uses wp_prepare_attachment_for_js to get all information about their images for their blade templates. Maybe this will help you to get started.

A maybe approach

Filter the_content on output. This solution with DOMDocument/DOMNode is heavy on the server and should be cached or saved or not done at all.

add_filter('the_content', 'my_image_filter');
function my_image_filter($content) {
    $doc = new DOMDocument();
    $doc->loadHTML($content);
    $images = $dom->getElementsByTagName('img');
    foreach ($images as $img) {
        $info = $img->getAttribute('alt');
        $alt = $doc->createElement('p', $info);
        //TODO: Adding $alt Node somewhere or replacing Image Node...
    }
    return $doc->saveHTML();
}

Another approach

While migrating the content, filter the alt tag and append some html after the images. If you have your images not within WordPress make a custom tinyMCE integration that handles images as described above.