What you will want to do is do a back up on your 5. I would advise turning iCloud and iMessage off, take your sim card out then wipe your 5. Do not re-insert your sim card in to your 5. Unplug your 5 which should now have no data on it. Put your sim card into your 6 (assuming you will be using the same sim card), then sync your 6 to your computer and restore from back up, this will transfer everything over, including your contacts and messages. Then turn iCloud and iMessages back on.
Job is done. Your 6 should have everything set up like your 5 was (effectively like you never changed phones) and your 5 should have nothing left on it and nothing attached to it and ready to send away to be recycled.