azjoe_02,
Since you are going outside you should spend the extra $ on a GFCI protected breaker. Here is a good link for connecting the new cable into your panel...
http://www.hammerzone.com/archives/e...er/install.htm
At the other end of the cable you will either need a weather-proof outlet box or a weather-proof junction box depending if there is a plug or bare wires coming out of the kiln. If its a plug just take a sketch of it to your local hardware store and pick up the matching outlet. If it has bare wires... the black or red should be hot (110v) the white is neutral and green or bare copper is ground. You can connect like colors from your feed cable.
If you have a connection strip inside the kiln and no wire coming out, you will have to make the connections there, you should send us a sketch of the connectors or a description of it and any markings next to the connection points.
Good luck, Jim
'Just a handyman trying to help'