Open Contacts on your computer and delete the contact.
Connect your phone to your computer, open iTunes on the computer and do a sync.
That should remove the same contact from your phone.
This method may differ depending on the OS of the computer. Doing an internet search may help.
Good luck with it.
I've been trying to
remove Groups of contacts from my iPad Mini, and I think I finally figured it out.
There is almost no data for this action using OSX Mavericks, because it is so new I suppose.