Get down to the dog's eye level before you take a single frame. Shooting from above makes a small dog look smaller and loses the connection that makes a pet portrait work.
Lens and settings
I use a wide aperture, somewhere around f/2 to f/2.8, and focus on the nearest eye. Animal eye autofocus helps a great deal, but check the focus point has landed on the eye and not the nose.
Dogs move quickly, so keep the shutter speed high. Around 1/500 of a second freezes a head tilt; faster still if they are running. Shoot in short bursts, since a head tilt lasts about a second.
Getting the look
A squeaky toy or a treat held just beside the lens gets the ears up and the eyes in the right place. Make the noise once, take the picture, then wait. The second squeak never works as well as the first.

Keep it short
Keep sessions short, bring treats, and stop while the dog is still enjoying it. Ten good minutes beat an hour of a tired dog.
