Here is my suggestion:
Just download this Comment Images plugin.
As of now it only support images. But upload work with all file types.
So just make a little tweak like this. So it will display attachment link.
Replace line 245 from
$comment->comment_content .= '<img src="' . $comment_image['url'] . '" alt="" />';
to
$comment->comment_content .= '<a href="'. $comment_image['url'] .'">Download attachment</a>';
But you mentioned, you want that function in theme instead of plugin. So if you really know what you are doing, just integrate that whole plugin code in your theme by changing file path.
Otherwise, just hire a programmer.