I believe the alt attribute of the image is stored in post_content, isn’t it?
To get the alt tag for an image with attachment ID $attachment_ID, you would just call:
$alt_text = get_post( $attachment_ID )->post_content;
To actually display the alt text dynamically on images that are already embedded in posts, you would have to do some heavy regex processing on the_content filter.
If you wanted to set it dynamically when adding the image to a post, I would add a filter on image_send_to_editor.