Basics
You know your pet's favorite places. Take him to one of those spots.
Don't force the pet to pose. The photo will turn out looking forced.
You may have to return to the places at different times of the day to use natural lighting for the best results.
Use your pet's favorite toy or treats to get his attention.
Position of Pet
Don't have the pet looking up at you. Go down to his level.
Take many shots of the pet's head, use a zoom lens if possible. Try to get the details of his eyes and face.
Don't hesitate to take numerous shots. Every one will be slightly different. This will give you a better chance of capturing the perfect pose for your pet's portrait.
Also be aware of the background of the photo. Try to not include the garbage cans, etc. This can be altered, but if possible try to capture what you want the portrait to look like.
Lighting
Try to use outdoor lighting coming from behind you.
Avoid using direct sunlight as this can diffuse the details of the pet.
A bright overcast day is perfect for your photo.
Using a flash can cause red eye and distract the pet.