You need to break the phrase into two clauses, the main and the dependent cause, and then connect them.
- The cat is clack: Η γάτα είναι μαύρη
- I am eating the cat: τρώω τη γάτα (or the soup or anything for that matter, just remember that this is the object in this clause and is in the accusative)
Connect them: Η γάτα, την οποία τρώω, είναι μαύρη.
This becomes obvious if you replace 'that' with 'whom': The .... whom I am eating/talking to/watching, is... Same thing really in English, with the relative pronoun that introduces the relative clause in the accusative. ;)