If you have not done so already, you can download TouchCopy from here.
This support article details how to print your SMS, MMS, iMessage and WhatsApp messages from directly within TouchCopy.
Firstly, Run TouchCopy by double clicking on the TouchCopy icon on your desktop. When the TouchCopy window opens, and you have successfully connect your device, select the "messages" tab for the left hand side of the TouchCopy application.
From here you should be able to see all of the messages contained on your iOS device. (1) Select the message conversation that you wish to print by select the corresponding contact's name. (2) Click the print button in the TouchCopy control bar to open your printer settings.
From here you can select the desired printer that you wish to use to print your messages, as well as other standard printer properties.