Translation:He was talking and looking at his own dog.
Not sure actually. English here allows you something that Czech does not like - the stray preposition. I am not sure the present sentence can mean that - but perhaps... really not sure. To be clear we would choose a different wording "Mluvil na vlastního psa a díval se na něj.".
"He spoke and watched his own dog" is also accepted. There is, however, a report in the system for "He spoke and watched AT his own dog" that came in at the same time as your comment, which may have been your complete answer. "Watched AT" doesn't work in English -- it should be either "looked at" or "watched."