There are no easy options for traveling by public transport between Lisbon and Cadiz.
There are expensive flights operated by TAP/Air Portugal between Lisbon and Seville from where you can get a bus or train to Seville.
Alternatively, you could take a long distance bus.
See the following article on my blog about travelling from Cadiz to Lisbon:
http://goo.gl/4BSHF
Just reverse the instructions and away you go.
