Usually the best photograph for a portrait is the most simple.
Lower camera to dogs level.
Photograph outdoors.
Keep background simple, uncluttered.
Hold your pets attention, looking up at you.
I prefer full face portraits or 3/4 views--with the eyes clearly defined, as most of your pets unique spirit is carried in the eyes. Make sure the photo is well lit, sharp, focused with a lot of detail in the face.
A good rule of thumb is the pet should take up 2/3rd of the picture.
Usually the best photograph for a portrait is the most simple.