"El perro va hacia el gato."
The pesonal "a" is used when you are using and direct object. In this case "Hacia el gato" is a place object (I don't the exact word in English in Spanish is called "complemento de lugar"). "to go" is an intransitive verb, it doesn't have D.O. because the verb has a full meaning.
You can't have two prepositions next to each other. It would either have to be "el perro va al gato" (but NOT because it's being treated as a personal 'a' – see caiser's reply – since this means "the dog goes to the cat") or "el perro va hacia el gato". Same as in English you can't say "the dog goes towards to the cat".