I see the problem you have asked is more toward an idea rather than any code so I also will only be giving you the idea to do so. No code will be provided by me on this context and its like your own job to do so also.
Here's the idea,
Like Button :
1. A Like button is a simple hyperlink
<a class="like_button" href="<?php echo $post_id; ?>">Like</a>
Now a Comment Box :
1. Again the concept is same just that this time, instead of an anchor tag, you will have an input type text and it will contain class and a name
<input type="text" name="<?php echo $post_id; ?>" class="comment_box" />
3. Now this time, you can store all the comments in form of a serialized array in a single column 'post_comment' in database's table 'posts'.
4. The steps are,
$result = mysql_query("SELECT post_comment FROM posts WHERE post_id = '$post_id'");
5. Unserialize the data received using the query operation. Before that, also check if the data is blank or not. Now you can append new comment like this
$array_comments = $_GET['comment_string'];
6. Now you have to serialize the updated array again and store it back to the database using update query.
Hope that this idea or walkthrough will help you achieve what you want to do. Please do mark it as solved and rate it.