How do I take apart my mousepad to clean it?

There is no easy answer that I know of short of taking it apart. The bad thing about the trackpad is that there are a lot of components that cannot be separated and then put back together afterwards without special tools. I would take the c720 apart as much as you can until you can get a clear view of the underside of the trackpad (if it is held in by glue do not pry apart leave as is only remove screws and possibly tape if you are confident you can reseal the tape). Get some alcohol and clean it with a microfiber cloth both under and on top. Give it an hour to completely dry (overkill, but better to be safe) and then try clicking while it is still taken apart. If it is better but not perfect repeat until it is acceptable. Otherwise try searching around for a replacement which is most likely going to be the entire bezel and not just the trackpad itself.