There are many ways to portray one's pets. The best way I found is to play with them, or when they are napping.
Step 1: Go for Funny
I found that my cat gets himself into some pretty humorous positions when he is stretching/napping. It can be a good candidate for a photo.
Step 2: Go for Cute
Photos + Cute Animals = Awesome. Pets often look cute. Take advantage of this.
Step 3: Play With Your Pets
Playing with your pets makes them look really cute a lot. Scratching them and/or giving them chew toys works too. (or have someone else do this while you snap the pics)
Step 4: Have Them Look Natural
Pictures of your pets doing natural things look good too. Follow them around for a bit. Wait for them to forget that you are even there.
Step 5: Get Close
Don't be afraid to put the lens close to your pets, often these make the best pictures.
Step 6: Be Creative
Above all, be creative. Don't be afraid to try crazy angles or take pictures of weird things about your pets. Also, certain colors complement your pets' coat. Experiment with this. (I got a little sunbeam crazy for the last few) Have Fun!