My cat has black mouth marks and bad breath. Is it cat flu?

Cat influenza is pretty rare, and your vet may have been referring to one of the more common viral diseases of cats that affects the upper respiratory tract and/or eyes, such as chylamidia, calicivirus, or herpes. Often these viral infections will produce secondary bacterial infections that respond to antibiotics, so this would definitely be indicated if it hasn't been tried. These viruses can also produce a condition in the mouth known as stomatitis, or severe inflammation of the gums and oral cavity, which often causes bad breath. Many times these cats need to have many teeth extracted in order to bring the condition into remission. I'd recommend seeing a specialist, or another vet who is familiar with these conditions and able to reach a definitive diagnosis and treatment plan. I would also recommend testing your cat for feline leukemia and feline AIDS, as these diseases can make cats more susceptible to other problems.