Q:

Where did my father serve?

Look, where I walk!
Listen, when I speak!
Babyface, together with a dog?
Skull, together with a cat and a mouse?
The first time I was a King.
The second time my mother was a Queen.
But I have also been the Joker.
In the future I will awake as a Chess champion again.
Where did my father serve?

A:

My current thought for the narrator:

 Mark Hamill

And thus, for the answer:

 Either The Empire, or the U.S. Navy, as that is where Hamill's father served.

"Look, where I walk!"

 "Look, walker in the sky" = "Luke Skywalker"

"Listen, when I speak!"

 Hamill has gained a reputation as a prolific voice actor.

"Babyface, together with a dog?"

 Hamill's role Babyface Boretti in the Scooby-Doo movies.

"Skull, together with cat and mouse!"

 In "Marvel Super Hero Squad: Comic Combat", Hamill did the voice of RED SKULL. Hamill did voice acting for th Tom & Jerry Kids episode "Droopy Man Returns".

"The first time, I was a king"

 Hamill's first appearance, according to IMDB, was as "King Blackyard / King Karuta" in "Exchange Student Zero"

"The second time my mother was a Queen."

 The mother of Luke Skywalker was Queen Padme Amidala.

"But I have also been the Joker"

 Hamill's role in the animated Batman series.

"In the future I will awake as a Chess champion again."

 In chess a champion is also referred to as a Grandmaster. 
 In the Star Wars expanded universe, Luke Skywalker becomes Grand Master of the Jedi Order. 
 Mark Hamill will be playing Luke Skywalker again in the new Star Wars film.

